Invariant Measures on Groups Citation Anderson, Douglas Robert (1956) Invariant Measures on Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/TPWC-6Z18. https://resolver.caltech.edu/CaltechETD:etd-02252004-103754 Abstract A generalization of A. Weil's notion of measurable group is formulated and various properties are developed. The properties obtained for the generalization are applied to a problem in the theory of group representations and to the characterization of separable topological groups possessing a measure. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Aeronautics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Bohnenblust, Henri Frederic Thesis Committee: Unknown, Unknown Defense Date: 1 January 1956 Record Number: CaltechETD:etd-02252004-103754 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-02252004-103754 DOI: 10.7907/TPWC-6Z18 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 749 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 27 Feb 2004 Last Modified: 10 Jul 2023 22:32 Thesis Files Preview PDF (Anderson_dr_1956.pdf) - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

INVARIANT MEASURES ON GROUPS Thesis by Douglas R. Anderson In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1956 ACKNOWLEDGEMENT I wish to thank Doctors Bohnenblust and Dye for their help and encouragement while the research for this dissertation was in progress. ABSTRACT A generalization of A. Weil's notion of measurable group is for- mulated and various properties are developed. The properties obtained for the generalization are applied to a problem in the theory of group representations and to the characterization of separable topological groups possessing a measure. it Il TABLE OF CONTENTS Introduction The Theory of Generalized Measurable Groups Application to Representation Theory Characterization of Separable Topological Groups Possessing a Measure 38 106 -1- INTRODUCTION This thesis deals with a generalization of A. Weil's notion of "measurable group" {'"'groupe mesuré"), (See (1) and (7).) In the first section various properties of this generalized structure are obtained, and it is shown how one can strengthen Weil's principal result on measurable groups. In the second section our generalized concept is used to find an extension of a result of Loomis in the theory of repre- sentations of locally compact groups. In the third and last section we apply the results of the first section to show how local compactness is a combination of measure-theoretic properties and completeness of the group uniform structure. Throughout this dissertation there is presumed a familiarity with the elementary topological notions and the approach to measure theory (1) expounded in Halmos' Measure Theory'’. Of course, a certain num- ber of terms which are either not well known or are taken here in a sense different from the usual have been defined in the body.of the pa- paer. Other notions and theorems are assumed known: e.g., the defi- nition of an integral (section 25 of (1)), the notion of a o-finite measure space (section 17 of (1)), the definition and existence of product mea- sure spaces, the version of the Fubini theorem in (1). Finally, I wish to express my gratitude to those who made this thesis possible. First, to Doctors Dye and Bohnenblust for their ad- vice, encouragement, and patience while my research was in progress. Then, to my sister Maria for her constant understanding and moral support. -~2- _I.. THEORY OF GENERALIZED MEASURABLE GROUPS 1. Definitions Definition 0: Given § and T, -rings of subsets of the abstract sets Sand T, respectively, we define S x T to be the O-ring generated by {mM x nha in Ss, N in cd) - Itis clear that given a finite collection of n measurable spaces \s,, 5} one may define inductively a O-ring n i=l ~< S., by the relation i=l : mt+tlo_ m _ _ XM SOK 8) Sn ar: i=l i=l If we make all identifications necessary to insure the associa- tivity of the process of forming Cartesian products of sets, the process of forming products of O-rings is associative. This will be formulated and proved as a lemma: Lemma: Let (S,S), (T, T), and (V, ¥) be measurable spaces, Then Proof: Defining R(E) to be the o-ring generated by the family of sets E and D x A to be the family of sets ir x alrep} > we make two preliminary observations. First, R(E x A) = (R(E)) x A. For, on the one hand, R(E x A)C(R(E)) x A. And, on the other hand, since R(E) C{k\K x AcR(E x a}, (R(E))x AC_R(E x A). Second, a Us = U Re, )). cel nel For, RU £,)cR( UW RE,)) cel «el cR(U RL E, )) ode] “et = R(U E, axel _ We can now prove the lemma. If (; {z, “el (Sx T)xV= a[U x ( fn x BlAcs, pet }) Jun] adel =, by the first observation, a{U a(faxp xE 7 A&B, 5et}| axel =, by the second observation, R(UfaxBxE, | Ac5, Bet] ih I R(fa «Bx ClAcS,BeT, Ce). By a similar argument, Sx (Tx V) R( {a xBxC lacs, BeT, cev}) . Hence, we have the desired result, Definition 1: Given a measurable space (X, S) we say that a real- valued function f on X (on Mé«S) is S-measurable (S-measurable on M) if for every & such that -0o 4 * £ + 0 fcletc) #0, £(x) és} eS. If itis clear in any context that only one measurable space is being considered then an S-measurable function will merely be referred to as measurable. Definition 2: Given a measurable space (X, S) we define a measur- ability - preserving transformation of (X, S) to be any one-one mapping, T, of X onto itself such that T(S) = S, "Thus for any E in S both T(E) and thE) belong to S. -4. Definition 3: We say that a measurability~preserving point- transformation, T, of a measure space (X,S,m) is non-singular if m(E) = 0 implies m(T(E)) = 0 for all Ein S, An important example of a non-singular transformation on a measure space is of course that of left-translation on the measure space consisting of a locally compact topological group, its 7-ring of Borel sets S, and the Haar measure on S. Definition 4: Given two-measure spaces (X,S,), (X,5,”), p is said to be equivalent to Vv (u~v) if p is absolutely continuous with respect to v , and vice versa, Definition 5: A @-ring G of subsets of a group G is called in- variant if the transformation S: (x, y) > (x, xy) on Gx Gis measurability-preserving relative to the product measur- able space (Gx G, Gx G). Definition 6: By a generalized measurable group [= coc] (G, G, m) we mean the structure composed of an abstract group G, an invariant oO ~ring G of subsets of G, anda non-zero, 9 -finite measure m on G, where left-translation by any member of Gis a non-singular, measura- bility-preserving transformation of (G,G,m). We call (G, G, m) bounded if G is in G (namely, if (G, G, m) is to- tally S-finite) and left-invariant if mis invariant under left-transla- tion. The definition of a GMG is of course suggested by that of a measurable group (a concept due to A. Weil}, which in our terminology is simply a left-invariant GMG. (The reader is referred to (1) and (6) -5- for a discussion of measurable groups.) And as with a measurable group an example of a GMG is afforded by a well known structure, namely, an arbitrary locally compact topological group G with the Haar measure on its o-ring of Baire sets, B. The proof that B is invariant in the sense of Definition 4 follows from the fact that the transformation | S: (x, y) ~ (x, xy) isa homeomorphism on G x G in the product topology, together with the fact that thefamily of Baire sets of Gx G coincides with B x B. (See Theorem 51.C of (1).) The use here of the o-ring of Baire sets of G in preference to the o-ring of Borel sets is partially motivated by the Lemma: Let G be a compact topological group, G the Borel ‘sets of G, and m Haar measure onG. If (G, G, m) is a measurable group, then the power of G is at most that of the continuum. (See p. 261, ex. 2 of (1).) Definition 7: Two GMG's (G, G, m),(H, H,n) are called (weakly or strongly) equivalent if there exists a measurability-preserving group isomorphism T of G onto H such that (weak case) mo pol n or (strong case) mo To! = an, aa positive real number. Thus each bounded GMG is weakly equivalent to a GMG in which m is finite. - Definition 8: The transformation R on G x G shall be defined by the relation: R(x, y) = (y, x). R is measure-preserving on (Gx G, GxG, mxm) as is clear from the definition of this space. ~6- Definition 9: Two functions, f and g, on a measure space (S, S,m) shall be said to be equal nearly everywhere (m) [= ne. (m)] if they are equal except on a subset of S which intersects every measurable set in a set of m-measure zero. A statement which is a point-function on (S,5,m) is said to be true for nearly all x (m) if it is true except for x in a set whose characteristic function equals zero n.e, (m). Proposition 0: If A and B are any two sets, then isrsya x B)] = xa Na and 1 1 1 Ay, yeB™ {s RS)(A x BY = d, ye(B")S. Proof: See Theorem 59.C of (1). Proposition l: Let G be a group and G an invariant o-ring of subsets of G. If (G, G, m) and (G,G,n) are GMG's, then S and sv! are non-singular measurability-preserving transformations on (GxG, GxG, mxn). Proof: For any M in G x G one has, by the Fubini Theorem, (m x n)(M) = 0 if and only if . n(M,) dm(x) = 0 and (m x n)(S(M)) = 0 if and only if n((S(M))_) dm(x) = 0. But (S(M)) = xM, and, (G, G,n) being a GMG, n(xM,) = 0 if and only if n(M,) = 0. Hence (m x n)(S(M)) = 0 if and only if (mx n)(M) = 0. If S is a measurability-preserving transformation on (GxG,- Gx G, m xn) then sg" has the same property; thus the conclu- sion of the preceding paragraph yields the result that S and sl are non- singular measurability-preserving transformations on (GxG, GxG, mxn). -7- Proposition 2: On any GMG, the transformations U and Ry. defined by the relations U =x—x 7! and Ry: x—> xy, are measurability-preserving and non-singular. Proof: Theorems 59.A, C, and D of (1), with an obvious modifi- cation in the proof of the latter theorem, carry over to the case of a GMG. The first two theorems are needed to prove the third, and the third is equivalent to the proposition under consideration, Corollary 1: The function f(x) = m(xEMF), E and F arbitrary members of G, is measurable for any GMG. Proof: By Proposition 0, [(s'rsyzxF7)] =x ENF. Since, by Proposition 1 above, S is a measurability-preserving trans- formation on any GMG and, by Proposition 2, Fl in G, g(x) = m({(s-'Rs)(E x Fy] = mx EMF) is a measurable function on (G,G,m) by Proposition 2. Hence, in view of Proposition 2, f(x) = a(x!) = m(xE{\F) is measurable on (G,G,m). Corollary 2: Any GMG, (G,G,m), has the property that if E.andF are in G and m(E)? 0 and m(F)> 0, then m({x|m(xEMF)> 0} )>0. Proof: {x|m(xE MF) o} eG by Corollary 1. Since § and sare non-singular on (Gx G, Gx G, mx m) by Proposition 1 and R has this property also, (m x m)((S"'RS)(E x F7'))> 0. Asa result, l 0¢ (mx m)((S7'RS)(E x F74) m({ [is"'rsye x F7)] ) dm (x) -8- = [etetsnr) dm (x) = m(x "EN F) dm (x) felm(c7 te NF)> 0} = fates NF) adm(x) ; felm(xE NF) > of 7 whence, in view of Proposition 2, m{( ixdm(xENF)> o} )>0. Corollary 3: Let (G, G, m) be a bounded GMG. Thenif hisa measurable function on every member of G and if h(xy) = h(y) n.e. (m) for each x in G, his constant n.e. (m) on G. Proof: Observe that g(x, y) = h(y) is measurable on (GxG, Gx G, m x m) as is also g(S(x,y)), in view of Proposition l. Since the hypothesis of this corollary implies that (Sx, y)) = g(x, y) n.e. (m) for each x in G, one can apply the Fubini theorem as follows: 0= | i lets, yd - (St, vl emt) d(x) E\F ~ / late, y) - g(Stx, yl seni) dm(y), where E and F are arbitrary members of G. From the arbitrariness possible in the choice of E and F, it can be concluded that for nearly ally inG, g(x, y) = g(S(x, y)) n-e. (m), i-e., remembering the definitions of g and S, h(y) = h(xy) n.e. (m). Thus there exists Yo in G such that -9. . h(xy,) = bly.) ne. (m), so that by Proposition 2 h(x) = hy.) nee. (m). The remaining results of this section, suggested by the well- known decomposition of an arbitrary locally compact topological group into the family of left cosets of a certain o-compact open subgroup, will be of fundamental importance in the remaining sections. These ‘results will allow us to prove a Radon-Nikodym type of theorem for a GMG and thereby make it possible to obtain results about an arbi- trary GMG which would otherwise be true only for a bounded GMG,. Lemma: Given a GMG, (G,G,m), Gis a topological space if the defining family, N, of neighborhoods of the identity is taken to be the collection of all sets of the form cl (cE NE)>Oo , 0<m(E)<oof and if the defining family of neighborhoods of.an arbitrary eiement x of Gis taken to be xN. Further, taking the defining family of neighbor- hoods of an arbitrary x in G to be Nx yields the same topology. For any Nin N, N= nol, and N is measurable and of positive measure. Proof: To show that we have a valid topology it is necessary only to prove that the neighborhood axioms are satisfied relative to the mem- bers of N, since for each x in G xN is the family of neighborhoods of x. First, we show that given M, N in N there exists K in N such that KCMIMN. Let M = {xlm(xE(ME)>0, 0<m(E)<o} , and N= fclm(xF MO F)>0, 0<m(F)<oo} . Observe that by Corollary 2 to Pro- position 2 there exists y, in G such that yo FO NE! has positive measure. vo FE is measurable by Proposition 2 and the non- singularity of left-translation. ) -LO- Asa result, | 0 <m(ENFy,) <0. But then, letting L = EfIFy,; K = fxlm(xL M1) >0} eN. Also remem- bering the non-singularity of right-translation asserted in Proposition 2, KClxlm(xE NE) of a ixlm(xFy, NFy,)>0} (x|m(xE (QE)> o} ft) {x [n(x flF)> 0} MIN. i In order to show that given any N in N and any Xo in N there exists inNa neighborhood of xX,» we prove first that given N in N and Zo in N there exists B in N such that Bz N, and we prove second that given N in N and Yo in G there exists M in N such that MCy,”'Ny,. Let N = fxlm(xENE)> of eN and Zo in N be given. Then define B= foe fra( x(z EN E)N (z= NE))> 0} . Since 2, «N and 0¢m(E)<«, BeN. And if xeB, m((xz, EME) m((xz, JENENxEMz E) = mix(z ENE) M2 ENE) 0. Then, by definition of N, Bz C N. Let N= fxlm(xE NE)> of eN and yoeN be given. Note that vo { xlm(xkE ME)> o} Yo Mt [e) a rw) [e) " xlm(ty xy,” VE NE)> d felmecty, 2) Vy, Ey) >0) by the non-singularity of left-translation. By the 7-finiteness of m there exists DCy,'E such that 0<m(D)<coo. Consequently, if m= fxlm(xDOp)>0} ef, then -Ll- MM Chlmtety, 8) Ny of = ys“ Ny... The two facts just obtained not only complete the proof that N is a true family of neighborhoods of the identity but also show that the same topology is formed regardless of whether a family of neigh- borhood at x is taken to be xN or Nx. By the first two corollaries to Proposition 2, any set of the form N= {xlm(xE NE)> 0, 0<m(E)< oo} , is measurable and of positive mea- sure. Finally, if N = fxlm(xE MN E)S0 § «N, then N = nol for xEME = x(x" E Nh E) and left translation is a non-singular measurability-pre- serving transformation on any GMG. It might be noted at this point that the topology constructed does not necessarily satisfy any separation axioms nor does this topology have, in general, the property that the group operation is continuous in it. The topology constructed on an arbitrary GMG in the preceding lemma will be called the m-topology on that GMG. It should be noted that if (G,G, m) and (G,G,n) are weakly equivalent, then the m-topology on G is equivaient to the n-topology. Proposition 3: In any GMG, ({G, G, m), there exists a subgroup H of G with the properties that il) HeG , 2) m(H)70, 3) His open in the m-topology on (G, G, m). Proof: Let N = Liclma(xE NE) >0, 0<mlE)<a} . Nis open in the m-topology, being a neighborhood of the identity in the m-topology, and, ~l2- as such, is measurable and of positive measure by the preceding lemma. Define the set n* for any positive integer k by the following re- lations: ly Nien: 2) NX = NEN, xpi. We shail prove by induction on k that for all k n* is open in the m-topology, measurable, and of positive measure. From the remarks already made about N = ni we see that itis only necessary to show the (+1) truth of our assertion for N to be implied by its truth for N , for any £> l. Assume N is open in the m-topology, measurable, and of positive (+1) | measure. Then it is immediate that N = wn is open in the m-topo- logy and, if measurable, of positive measure by Proposition 2. It re- net) mains to prove that is measurable. Note that xN nn’ 4 é implies that there exists x, in G such that for some M = fxlm(xF MF)> 0, 0<m(F)<0o} x MCxN ane ’ since xN is a neighborhood of x in the m-topology and N” is assumed to be open in the m-topology. From the last lemma proved and the non- singularity of left-translation we conclude that x M has positive mea- sure. xN is measurable and N is assumed to be measurable. Thus xN{|N° is measurable and of positive measure for all x such that anne #¢, or, Sad. Hence, the last lemma proved having shown N = nol foclan(xN ONY) > o} = Lela ON -13- | NN f NWN _ yu. " foclm ta N4> 0} iT Since N and, by the inductive hypothesis, nare in G, then Corollary 2 to Proposition 2 applies, yielding the result that éxlm(«xN nw hyo! , and thus Nis in G. This completes the proof that n* is open in the m-topology, mea- surable, and of positive measure for all positive integers k. Consider H = CJ n*, On the one hand, we have, from what we have just shown, that His measurable, of positive measure, and open in the m-topology. On the other hand, His a subgroup of G. First, by the last lemma proved above, we see that N = no! and so n* = onky7!, Next, nin? = Nuacd Finally, N contains the identity of G. Thus we have exhibited a subgroup of G which is measurable, of positive measure, and open in the m-topology, thereby proving this proposition. Proposition 4: Given a GMG,(G, G, m), let H be a subgroup, mea- surable and of positive measure. Let E in G have positive measure. Then E intersects in a set of positive measure at most a countable number of cosets of H, say {x,H] im D where i 7 j = x,HOx,H = ¢. Further, m(E - U, (E)x,H)) = 0. Proof: Let fo} be the family of ail left cosets of H. By 00 -O-finiteness E = LJ A, where 0<m(A,)<oo. Since the members of i=l } Cid are mutually disjoint measurable sets, so also are the mem- bers: fac, Lf m(A,/) C,,)70 for a more than countable collec- el’ -14- tion of ‘a's, there exists 570 and an infinite collection of «'s for which ~m(A,C,)>8, which in view of the fact that ©, Cy = ¢ for a # B would imply m(A,) =o, a contradiction. So let A = to} ;- 1 be the countable collection of «'s such that «in A implies m(C 1) A,)70 for at least one i. Then m(E MC,)>0 if and only if x is in A, aroving the first part of this proposition. Now consider the measurable set F=E- Ci (EMG, ). j=l j 00 This set, by the properties of ic} just obtained, intersects jJj=l every Cy in a set of measure zero. Observe that, by Proposition 0, -1 -ly. -1 x HOF = ls RS)(H x F de. for each xin G. On the basis of this one sees that every set of the form -1 -1 is measurable and of measure zero. Then, as 1) Hx PlisGxG~ measurable, 2) (s7'rsy7! = srs is by Proposition 1 a non-singu- lar measurability-preserving transformation on (GxG, GxG, mx m), 3) AGx G-measurable set $ has (m x m)-measure zero if and only if m(S,) = One. (m), or m(S”) = 0 nee. (m), 1 we have that(m x n(Hx F™)= 0. It follows by Proposition 2 that m(F) = 0, which proves the remainder of the present proposition, -15- Definition 10: A measure space (S,S,m) is said to be complete _ if and only if for every FeS such that m(F) = 0 EeS whenever ECF. -Definition ll: Let (S, 5S, m) be an arbitrary measure space. Then s'™) 2{keslk = (E - N,)UN,, EeS,3Msm(M) = 0 and NU N,cMé is a o-ring containing s, and the measure m on gm) defined by the equation m((E - NUN m(E) 2) = ; -— * « =(m) : agrees with monS. (The complete measure space (S,S ,m) is called the completion of (S, Ss, m).) Proposition 5: Let (G,G, m) and (G, G,n) be a pair of GMG's such that n is absolutely continuous with respect to m and (G, G, m) is a complete measure space. Then there is a non-negative function f on G, measurable on every set in G, such that for every E in G n(E) = f f(x) dm(x) E If there exists a function g on G with the same properties as f, then f=gn.e. (m). Proof: By Proposition 4 we can dissect any E in G into a set N of m-measure zero and the collection {E. Av =] of the intersections of E with a countable collection of left cosets of a subgroup H in G such That M is uniquely defined is shown as follows. Let K in sim) be such that K =({E - N,)UN, = (D-R)UR,, where D and E belong to S and N,U N,UR, UR, is contained ina member N of S having zero m-measure, Then, E-NC(E.- N,)UN,CEUN, and D-Nc(D-R ani DUN. Since m(N) = 0 and (E - NUN, = R,)UR,, m(E)sm(D), and m(D)<m(E), i.e. m(D) = m(E). Thos, w(K) if is uniquely defined. -16- that m(H)>0. . Let ica}, be the family of all mutually disjoint left cosets of the subgroup in question. Then we know by the Radon-Nikodym Theorem (valid for totally 7-finite measure spaces) that for every there is a non-negative function h, on G (vanishing outside C,) such that 1). : —alFNC,) | hy, (x) dim(x) FNC, [ hy (x) dm(x) F for every measurable F and 2) if k, has the same properties as h, k, =hyn.e. (m). Thus, because of the already-mentioned decomposition of the arbitrary measurable set E, re n(E) = > n(E,) i=l = 2 n({EM Cc.) i= i [oe] . = > hy (x) dm(x) , i=l “E-N i n er) and, since [S : h. : is an increasing sequence of non-negative i= von: measurable functions, rr / 00 > hy, (x) dm(x) = ( > hy (x)) dm(x). i1=1 E-N i - t=l i So let us define f on G by the relation: f(x) = he (x) for x in.C, . Then, since hy (x) = Kk (x) hy (x) 1 xs i -L7- for all x and oo . oO rahe - U fc, \ - m(N) i=i i = 0, f is measurable on E by the completeness of (G, G, m), and the formula just obtained becomes: n(E) = J £(x) dm(x). E-N Inasmuch as m(N) = 0, we have that n(E) = f(x) dm(x). E Also, if g is a function on G with the same properties as f, eX, isa ae function having the same properties as hy so that tke, = hy nee. (m). But, by definition, so that eh -«X n.e. (m). CK CK " Since ® was arbitrarily chosen and since every set is contained ina countable collection of CL is if we delete a set of measure zero, g=fn.e. (m). — n Proposition 6: If lic, G,, m,)} is a collection of GMG's, i= 1 then the assertion of Proposition 5 is valid for the completion of mea- sure spaces of the form n , n n Proof: We can prove by induction that x< G, is the same as i Al =1 the 7-ring generated by la, KAD oeeee ACO c,lajec, - This -18- combined with Proposition 4 yields the ‘result that for every Ke >< G; i=l _ there exists Ne x G, for which (m, xX M5 Xeeee. XM nN) = = 0 and K - NCO “J (3, Oy K eeees xa,™u ); here H, is a subgroup of G, of the sort. constructed by Proposition 4. Regarding x< G; as a direct- isl product group, we see that Hy Keeeee X A is a measurable subgroup of x G; having positive measure and that aH, Keeeee XA ny is i=l always a left coset of Hy Xe.eee. XH anda set of positive measure. n (m, x ..... x m_) Hence, by the definition of b< a] } n n (m,x...xm)i=l Ee (>< G) there is an “3 in the same O-ring such is that (m, Keecoee XM n)(N) = 0 and E - nel J (a; (l a, K eeeee «aH, ). j=l This last conclusion makes it possible to repeat the argument of , for every Proposition 5; so, the assertion of the present proposition must follow. Lemma: Let (S, 8) and (T, T) be measurable spaces. Suppose ACS and BCT. Then (A x B)NS x T = (ANS) x (BAT). Proof: We observe first that (Ax B)NSx T contains all of the . collection of sets which generates (AM5) x (BVT), namely, fe xFCSxT | Ee(ANS), Fe(B ni} . This shows that = (ANS) x (B ADC (A x B)NS x FT. Now, by the same reasoning as was used to get the preceding, R, = (AMS) x (BSN T)C(A x BY)NS x T, R, = (ANS) x (BN T)C(AS x B)NS x T, and Ry = (ASS) x (BSN T) CAS x BS) MS x T. Thus, td fs, CSx TIER, i=l, ..., bcsnF. i=l -19- Since all the R,'s are 7 -rings, since no two R,'s have any set in com- mon, and since R contains every member of the family of sets gener- ating Sx T, Risa O-ring containing the generators of § x T, which means that As a result, We see now that any member E of (A x B)MS x Thnot belonging to R = (ANS) x (BN T) must, being a member of (A x B)/) R, contain a non-void subset of {A x B)*. But this contradicts the assumption that E is a subset of Ax B. Thus (A x B)NG x TD(AN8) x (BN T), which, in view of the fact that the latter family contains the former, proves that (A x B)) (Sx T)=(ANS) x (BN T). Proposition 7: Given two o -finite measure spaces, (S, Ss, m) and ‘ (T, T, n), consider (Sx T, (Sx Tim * n) m xn), the completion of yim x n) (Sx T, Sx T, mxn). Letf bea non-negative, (Sx T -measur- able function on Sx T. Then, except for x ina set of m-measure zero, f(x,y) is a T™) measurable function of y, and, except for y ina set of n-measure zero, f(x, y) is §(™) _measurable function of x. Further, pix, y) dn(y) _ is 3'™) measurable, if we define it in some arbitrary manner at all x for which f(x, y) is not 7) _,easurable, and, with a similar qualifica- tion, | Leen dm(x) -20- is S m)_ measurable. Finally, [ fey) de AY, nH [ fle, y) am(y)) ale) Sx T a Ss T Lf en am(s)) diy). tf Ss Proof: Let us note first that by the assumption of Sx t)™ x n)_ yim xn) there must be measurability for f and the definition of (Sx T) a set E in Sx T such that (x, yleS x T1its, y)>0}CE. Then, since the o-ring {Cs x T| 3FeS, GeTaKCF x ch contains Sx T, there must be an AeS and a BeT for which { (x, y)eS x Tl£(x, y)>0h} CA x B. In addition, f£, considered as a function on Ax B, is a non-negative x n) function measurable relative to (A x B)f} (S x rl oa , which, using ) (m x n) the lemma just proved, is identical with (afYs) x (BIT . But, A and B each being the union of a sequence of sets of finite measure, we can apply Theorem 9.10 of (5) to f considered as a non-négative, = — ,(mxn) . measurable function on (A x B, ((afys) x (Bf) T) ) »> mxn). Hence, remembering that (A{]S) x (Bf) T) (m x n)_ ‘ (a x BS =P), APS = as!™, ana BITT)™ = BI] 7m) and that f vanishes on (A x B)*, the assertion of this propo- sition follows. We now have the tools necessary to prove some uniqueness pro- perties of GMG's and to discuss the existence of invariant measures on the o-ring of a GMG. -21- 3. Uniqueness Theorems. In this section we shall discuss certain uniqueness properties of GMG's. We show that the family of null sets ina GMG is dependent only on the underlying measurable space and that the left-invariant GMG corresponding to the measurable space (G, G) is essentially unique if it exists. Proposition 8: If two GMG's have the same underlying measur- able space (G,G), they are weakly equivalent. Proof: It follows from the definition of the transformation R and the assumptions on S that s7'rs is a measurability-preserving transformation on (Gx G, Gx G, mxn), if we consider two GMG's of the form (G,G,m), (G,G,n). Further, by Proposition 0 [is "RSA x BY. =x ats and Ay”, yeB™! [is TRS ya x BY = { “ac? 4, yeB where A and B are arbitrary members of G. Asa result the following ‘formula is true: frecta Bo) amex) = ff, m(ay™') dnty) - B Thus, since m and n are assumed to be left non-singular and since by Proposition 2 the inverse of set of m-measure zero has m-measure . zero, we can see that if n(A) = 0, then m(A) = 0. Interchange of the roles of m and n in the preceding argument yields the converse result and completes this proposition, Theorem 1: Two left-invariant GMG's having the same underlying measurable space {G, G) are strongly equivalent. ~22- Proof: Let the two left-invariant GMG's be denoted by (G, G, m), (G, G,n). By Proposition 8, they and their completions are weakly equivalent. Also, alm) = am) By Proposition 5 there exists a non-negative function h on G such that h is G'")_measurable on every am) -measurable set and M(B) = f hy) any) for each E in qm), As a result, we have for any E in qin) and any x in G, {nly aBty) = CE) = mee) = f E nly) any) =f n(xy) ay). Therefore h(xy) = h{y) n.e. (fH), so that by Proposition 4 and Corollary 3 to Proposition 2 nly} = constant-n.e. (n).: Further, the. assumptions that nd 0 and that m. is: not identically-zero make the constant positive. So we have that for any E inG (E) = m(E) = h(x) dni(x) = (k) dii(x) = m m f n(x f k* f X (x) dni(x) =k* H(E) =k. n(E), E where k>0. This proves the strong equivalence of (G,G, m) and (G,G,n). It may be remarked that the preceding provides a new proof of the uniqueness of the Haar measure ona locally compact topological group. Forif Gisa locally compact group, G its family of Baire sets, | and m the Haar measure on G, then as shown in Section 1 above (G, G, m) is a left-invariant GMG. Theorem | now enables us to say that m is unique up to a multiplicative constant. Since, finally, Theorem 52.H of (1) states that for a locally compact space agreement of two regular Borel measures on all Baire sets implies agreement on all Borel sets, ~23- the Haar measure on a locally compact group is seen to be unique to within a multiplicative constant. 3. Existence Theorems ‘The present section is devoted to showing that any GMG is weakly equivalent to a left-invariant GMG. Since left-translation is a non- singular, measurability-preserving transformation on a GMG, (G, G, m), we may apply Proposition 5 to (G, aim) ,m), the completion of (G,G, m), to obtain a non-negative function f(x, y), measurable in y on every set in qi™) for each x and having the property that meee) = f fx, y) amy) for every E in aim), Speaking intuitively, f(x, y) is the density of the measure m at x relative to y. So, if for some fixed y, say Yo: f(x, y) ) were measurable on every set in qim , the measure on Elm) defined for each E in Gm) by the formula, _ 1 = n{E) = [ t. y,) dm(xy,), ‘would appear to be a left-invariant measure equivalent to ma. nis ‘a fortiori equivalent to mon G. We shall show that this argument, together with the assumption about the measurability properties of f(x, y), is essentially correct. We begin with some definitions. Consider a GMG, (G, G, m), and _ the complete measure space (G, Gi™) m). If m,(E) = m(xE), then d(m,) f(x, y) = dm If (mx m)(F) = (mx m)(S(F)), then -24- d (mM). d(x m) Lemma: Consider (G,G,m), a GMG. If £(x,, 220 X, )is n \(mx...xm) <1 (mx. ..xm) ( 4 c) - measurable on every set in{ X < | i=l i= 1 so also are E(x) +++, Ki, 2205 X)) and f(x, oe X 52 sees %)) , i<j. Proof: Since the transformation S is non-singular (by Proposition 1) and measurability-preserving on (G x G, G x G) and since the process of forming products of o-rings is associative, the transformation on ( x G, x G) defined by: i=l i=l U: (Kyo eee Kp ee ee Kye eee X,) Oe, peers Xs *5, Riggers ey pap tt? ¥y ) (x), eee; S{x,, *;), Kipper 1X5 Xiay ap oe +X) (Xp, 606 Ki, 200s X:Kiye00,X%)) ad is 1-1, non-singular, and measurability-preserving. Because of the nonsingularity of U, it is then a measurability-preserving transforma- me . »xm) tion on ne G,.( N G, , IX..eeM). £(U(x,,...,%,)) = i=l -1 fle vee mmp eve, ) and £(U~ lac eee x nd) = fq ee Ky Xi, 000, %))- J - Hence the lemma. Lemma: Let (G, G, m) be a GMG. Then there exists a function hon GxGxG such that a) for every x inG f(x, z) = h(x, y,z) n.e. (m xm), b) his (GxGx Gy ™ *mxm) easurable on every member of (GxGx Gq) * m x m) a c) for every x inG h(x,y,z)is (Gx G)™ * m) _mea- surable on (G x Gym xm), -25- Proof: Let x be a fixed, arbitrary element of G, and let E, FCG ‘ be arbitrary measurable subsets of G. First, note that m(yF) = m((S(E x F)) hence, m(xyF) is a G- measurable function of y on E, left-translation being measurability- \ preserving. Then, keeping Proposition 5 in mind, / m(xyF) dm(y) _f I/, £(x, 2) ote) dm(y) E E yF Sf [/ Rca x FY 2) £0s, 2) sata dm{y) S(E x F) if Fy, z) = f(x,z), a function G x aim * m)_measurable on every set inGx aim xm) for each x, © -/ F (y,2) d(x m)(y, 2) ExF S(E x F) _f F (S(y, z)) d(m x m)(S(y, 2) ExF -[ f(x, yz) d(m x m)(S(y, z)) ExF Second, we have that for any K in G, remembering Proposition 7, [ m(yF) amily) = [ oa [sos x Fy] .) amty) K G y = (m x m)(S(K x F)) -26- = (Mx m)(S(K x F)) KxF -{ ([ e0-2 am(e) amy). K gly, z) d(m = m)(y, 2) Thus, since frimeyr) 4 f stey, 2) amte)} _ 7! = = x"'fy mlyF) 7 f gly. 2) am(z)}, we have foreachx, m(xyF) = f g(xy, z) dm(z) n.e. (m). ro. Consequently, for each x f m(xyF) dmM(y) = (J g(xy, z) amte) Jam). E E\ F Then, since the assumption that left-translation is a measurability- preserving, non-singular transformation of (G, G, m) makes (m x m) Gly, z) = g(xy, z) (G x G) -measurable on every set of (Gx Gym * ™) Proposition 7 applies to Gly, z). Hence, f[ mleyF) amy) = f (/ g(xy, z) am(e) ) amy E E\ F -f _ glxy,2) GE My, 2) ExF The preceding paragraphs have shown that the measures defined on (Gx Gi x m) for each x in G by kK ,(M) = J f(x, yz)g(y, z) d(mx m){y, z) and (M) = J glxy, 2) (me m)y, 2) -27%- will agree on the ring R of all unions of finite, mutually disjoint col- _lections of measurable rectangles in (G x G)™ * m) We note now that R generates Gx Gand that, by Theorem 13.A of (1), that two measures coinciding on a ring and defined on the 7-ring generated by that wing coincide on the 9-ring. Hence for every x in G and every Min GxG, dh (M) =_(M). But it is immediate that for-every x inG d.. = K on (Gx q)™ * m) | So, for each x inG f(x, yz) = g(xy, z)/gly,z)n.e. (mx). Here M(x, y, z) = g(xy, z)/gly, z) is (@xGx6G)™ xm x ™m) measurable on each member of (Gx Gx q)™ * m x m) by the definition of g(y, z) and the preceding lemma, and for each x k(x, y, z) is (G x G)™ x m)_ measurable on each member of (Gx Gm * m) Now for each fixed x inG s7' Ey, 2)l £0, ye) ¢ kets, y, 2} = fy, 2)l £0, 2) Ae by, ya) and s7! is a non-singular measurability-preserving transformation on (GxG, (Gx Gym * m) mxm). So, for eachxinG | f(x, 2) = K(x, y,y 2) nee. (EM), and for each x k(x, y, y7tz) is (Gx Gym * ™m) measurable on every set in (G x G)™ * ™m) by the (Gx G)™ * m)_measurability property of K (x, y, z) and the preceding lemma. Also, K(x, y, y tz) is measurable. Lemma: If his the function of the preceding lemma, then a) for every band xinG L. _ h(b, y, xz) n (aT) ht, y> Zz) ~ h(bx, y; Z) 7S meee b) for every bin G h(b, y,xz) and h(bx, y, z) are =o (GxGx Gym * mx m) measurable functions of (x, y, z) on ~28- every member of (Gxaxa)™ xm xm) Proof: Using ‘Proposition 7, the fact that left-translation =(m) . preserves rol .“smeagurability, and the last two lemmas,we can per- form the following computation, If E and F are arbitrary members of G, then [ _alox,y,2) a(eremyy,2) = f —_flox,z) da My, 2) ExF ExF -f AL £ (bx, 2) d7A(z) amity) - f m(bxF) dm(y) E -S(L. f(b, z) amm(z) ) amy) E “S(L ff £00, x2) (6, 2) am(e)) am(y) =f _ fb, x2) f(x, 2) a(erem)ly, 2) ExF = f h(b, y, xz) h(x, y, z) d(mx™m)(y, z). Ex F But this means that if for every V in G x G we define r(v) = f h(x, y, 2) dex FA)y, 2) Vv and s(V) = f ne, y,xz) h(x,y, z) da m)(y, z) then r and s are measures of G x G which agree on all measurable rectangles inGxG. Then, using the same reasoning as in the pre- ceding lemma, we can conclude that r = s on all of Gx G and, indeed, on all of (Gx G)™ * m) Hence, for every b and xinG h(bx, y, z) = h(b, y, xz) h(x, y,z) n.e. (mx™), or, what is the same, for every b and xinG -~29- 1 _ hb, y, xz) h(x, y, 2) h(bx, y,z) n.e. (Xm). It follows from the last two lemmas and the fact that left-transla- tion is non-singular and G-measurability-preserving that h(b, y, xz) and h(bx, y, z) have the asserted measurability properties. | Lemma: The transformation T on Gx Gx G defined by the equa- tion | T(x, y, 2) = (xz, y, 2) isa(GxGx ay xm x m) _measurability-preserving transformation, Both T and to! are non-singular transformations on (GxGxG, (GxGxG)™*™*™) aemem). Proof: We note as a preliminary that if we define the transfor- mations L, M and N as follows they are (Gx Gx Gym xin x ™) _mea- surability-preserving and they, together with their inverses, are non- singular on (Gx GxG,(GxGx G)™ * me ™) m x mx m)? a) Lix,yy2) = G4 y 27), b) M(x, y, 2) = (2, y, x), c) N(x, y,z) = (x, y, xz). To begin with, the mapping which sends x onto xt for every x in G is non-singular and measurability-preserving on (G, G, m); whence, by the Fubini Theorem and the definition of GxGxG, Land Lb are non-singular and measurability-preserving on (GxGxG, GxGx G, mxmxm). It follows that they have the same properties relative to (Gx GxG, (G xGx G)™ x m x m) ,mxmxm). . .,As noted after Definition 8, the mapping R on (GxG,Gx G, m x m) such that R(x, y) = (y, x) is a measure-preserving ~30- . transformation; thug it follows by definition of (Gx Gx qm * m x m) andmxmxm that MandM are measure-preserving transforma- yim xm xm) 2 ae), tions of (GxGxG,(GxGxG _ Finally, let us use the fact, noted in the last paragraph, that the transformation (x, y) — (y, x) is a measure-preserving transformation on (Gx G,G x G,mxm), and let us remember that the transformation (x, y) > (x, xy), as well as its inverse, is non-singular and measurability-preserving on (Gx G,GxG,mxm). Then, by the definition of (GxGxa)y™ xm x m™) the transformation N is (GxGxay™ mx m)_ measurability-preserving, and both N and nv! are non-singular. A computation shows us that the transformation T is expressible by the formula 1, -1l T=LMNM Ls; hence, Tis (GxGx Gym * ™m x m) -measurability-preserving, and ‘both T and T7! are non-singular. Theorem 2: Givena GMG, (G, G, m), there exists a measure u on G such that (G,G, ») is a left-inva riant GMG and (G, G, ») is weakly equivalent to (G,G,m). If (G x GxG, (GxGxG), m x m x m) is the completion of (GxGxG, Gx Gx G, m x m x m) considered as a measure space and if h and T are, respectively, the function and trans- formation of the preceding lemmas, then for every E inG w(E) = f RU 2) EE), yz) Ex Fx K, (Here Fy and K, are fixed sets of positive, finite measure belonging to -31- O Proof: By the preceding three lemmas and Proposition 7 wt) is positive and finite n.e. (mx mx m) and is (GxGxaqym=emem)_ measurable on any member of (GxGxa)y™ * mx m) Thus » is well defined for all E in G and determines a o-finite measure on G. Thus (G,G,y) is a GMG weakly equivalent to (G, G,m), as a result of the definition of a GMG and Proposition 8. We now complete the proof by showing } to be left-translation invariant. Let a be an arbitrary member of G and E be any member of G. Then, 1 —— w(aE) = RT jy, 2) d(m= * m){x, y, z) lee) er ex, A(T", y, 2)) aad 1 x Fy x Ko (J f — arts) Jomty) ate e) ak) (Jf M eet) a(x) ) am) ) amt ( f f(a, x) h(a, 2 x) —— _ (f x Fy * gna, d(m xm) (x, ”) dmi(z) wee d(m x mx m)(x,y,z) . ExF xk _ Dlax2"y,2) o fo) The steps of the preceding computation are justified by the results f “h, “Ie “he =f of the last three lemmas plus Propositions 5 and 7, except for the im- (m x m x m) plicit assertion that h(axz7|, y,z) is (GxGx G) -measurable on E x Fy x K, since h(xz7) y,Z) has the same property on -32- (aE) x F, x K.- “But by Proposition 2 the transformations | x. ax | and Ka x are noh-singular and measurability-preserving on (G, G, m); whence, (x, y, 2) — (ax, y, 2) and (x,y,z) = (ax, y, 2) are non-singular and measurability-preserving on (GxGxG,GxGxG,mxmxm), which means that (x, y, Z) — (ax, y, 2) is measurability-preserving on (GxGxG,(@xGx Gm *™*™ omen). Now let us examine the formula just obtained for p(aE). If h(a, y, XZ) q,(y.2) = Day x2 h(ax, y, z) we know on the one hand that for all x 1 ——_ hk, yz) = q, &, y,z)n.e. (mx ™m) ata and we know on the other hand that -1 h q(T" x, y,2)) = B&D Further, we know by the next to last lemma above that TTA and mxmxm yf )_measurable on every member of g(x, y,z)are(GxGxG (Gx Gx Gm “me m) so that by Proposition 7 1 _ — it ysz) = q,(. y.2) nee. (mx mx m). -~33- But these facts and the fact that T and To! are non-singular, mea- surability-preserving transformations on (GxGxG,(GxGxaG)'™* ™*™) mx mx m) yield the following: h -1 BOY) 2g (Te, y, 2) _h{axz yy, 2) 1 = —_—___—_—. n.e. (M x m x m) ~ (Tx, y, 2)) As a result, C Bla, 2) d(m@ Xm ¥ M)(x, y, z) f h(axz ), ¥> z) re) ° _ —-—— (m x m x m)(x, y, 2) f h(T™ (x, y, 2)) ExF xK fe) o Thus, remembering our formula above for u(aE), p{aE) = p(E). This establishes the left-translation invariance of p and concludes our proof. Finally, we wish to show that our results yield a generalization of a theorem of A. Weil on left-invariant GMG's. In order to discuss this theorem, and,for later use in Section III of this thesis, we must state certain definitions from the theory of uniform spaces. (See (8).) - Definition A: Given an abstract set E, let C and D be any two sub- sets of Ex E, the cartesian product of E with itself. Then we mean by CD the set of all (p, r) in E x E such that there exists a q in E for which -34- we have both (p,q) in C and (q,r) in D. Further, by co! we mean the set of all (p,q) in Ex E such that (q, p) belongs to C. Definition B: Given Ex E, the cartesian product of an abstract set E with itself, we shall call a family {Va} cer of subsets of Ex E a system of neighborhoods of the diagonal if and only if the following properties hold: uy Av =A={(p,qeEx Ep =al, ael 2) For every pair, @ and $8, in I, there exists a y in I for which VL CV, [1Ve > y¥—Va (Vp 3) For every « in I there is a 6 in I such that -1 VA(V CV_ .- Every member of {igh ael shail be called a neighborhood of the diagonal. An example is afforded by any topological group G. Let {u} yed bea defining system of neighborhoods of the identity for G. Then {hex is a system of neighborhoods of the diagonal if we define for every dN: Wy = {(e,a)eG x G gepUy}. Definition C: Given a set E, let there be defined a system of neighborhoods of the diagonal Wabnet for Ex E., Then E is a Hausdorff space if we assign to every p in E the family of neighborhoods (Va) baer" (See p. 8 of (8).) By (Va), we mean, in accordance with the definition of Section 34 of (1), {acz (p, ae Vogt. We define the couple (E, {vat oer) to be a uniform space, and, for the sake of simplicity, we make the convention of referring to (E, {Va} oer) simply as 'E", Definition D: If (E, { Vohaer) is a uniform space, a family {coh scn of subsets of E is said to be a Cauchy filter if -35. 1) exery subset of E containing a C, is a member of ) | § {ce} sex - 2) the intersection of every finite subfamily of {c,} bek is non-void and belongs to{c,} Sek? 3) for every @in I there is a C, such that ) Cs x C.CV4 . {c,} Sek is said to converge if there is a point p in E such that the family of neighborhoods { (Valo baer is a subfamily of {c,} Bek ° {cr} eo is then said to converge to p. Definition E: A uniform space E is said to be complete if every Cauchy filter made up of subsets of E converges. Lemma: Given a uniform space (E, { Vat ex) there exists a complete uniform space (E, {¥a} Gel? such that E is equivalent toa dense subset E!' of E in the following way: there is a biunique mapping £ of E onto E! such that (f x f)(V,) = (E'x E')(\Vq for alla. Eis unique in the sense that any other uniform space with the same proper- ties has the form (g(E), {tg x eV, )} where g is a biunique mapping ae”? of E. (See Theorem 2 of (8).) We can now state the theorem which we desire to generalize: Theorem 3: Let (G,G,_m) be a left-invariant GMG. Then G can be given a topology T which can be defined by requiring a family of neighborhoods of an arbitrary element x to be the family xU of all sets of the form: x{ ylm((yE - E)UCE - yE))<e, 0<e <2m(E}<oo} ; all members of xU have positive measure and, when x is the identity, are symmetric. The subcollection xN, consisting of the members of xU for which m(E7}) is finite, has the properties that all its members -36- have finite measure and that itis an equivalent family of neighborhoods of x. Special properties of T are: a) for any neighborhood U of the identity there exists a neighborhood V of the identity such that vv CU; ‘b) given any neighborhood V of the identity and any a inG there is a neighborhood W such that awa ‘cv; c) there exists a neighborhood of the identity coverable by a finite number of left-translates of any other neighbor- hood of the identity. Suppose there is made an additional hypothesis on (G, G, m), namely, that for every a in G there exists an E in G such that m(aE NE)<m(E). Then the topology T has the Hausdorff property, so that G is a topological group in T. Finally, the completion G of G is a locally compact topological group such that if G is the family of Baire sets and 4A is the Haar measure on G then G{|GCG and AE) = m(E(\G) for all EinG. It might be noted that this is a combination of the results of Sec- tion 62 of (1) and the remark following Theorem 10 of (7). Our generalization is: Theorem 4: If (G, G, m) is an arbitrary GMG, then there exists a unique equivalent left-invariant measure n such that the first paragraph of conclusions of Theorem 3 holds for (G,G,n). Further, if for every a in G there exists an E in G such that m(aE QE)< m(E), then the second _ paragraph of conclusions of Theorem 3 holds for G. -37- Proof: The first assertion of this theorem follows from Theo- rems | and 2 together with the first paragraph of conclusions of Theo- rem 3, The second assertion follows from the equivalence of m ton and the second paragraph of conclusions of Theorem 3. It can be seen that this last theorem is a true generalization of A. Weil's theorem, for the uniqueness of n means that when m is left invariant m(E) is positive and finite if and only if n(E) has the same two properties; whence, if the members of a base at the identity for the topology T all have positive, finite n-measure, then they all have positive finite m-measure. This takes care of the only point in which the generalization is not apparent. -~38- I. APPLICATIONS TO REPRESENTATION THEORY The purpose of the present section is to prove a generalization of the succeeding theorem of L. H. Loomis: THEOREM: Let p—>U, be a strongly continuous representation of alocally compact group G by unitary operators ona hilbert space H, and let A->P, bea non-zero Boolean @-~homomorphism from the Baire sets of G onto an Abelian family of projections on H, such that uOP A= PoaU, for every p in G and every Baire set A. Suppose first that His ir- reducible under the combined families fu, and ip » Then there exists a unique unitary mapping of L(G) onto H such that U, corresponds to left translation through p, and Pa corresponds to multiplication by the characteristic function by of A. That is, TU TE (s) = £(p 7's), and Tp re (s) =¢, (s) £(s) « A A ‘In general H is a direct sum of subspaces each of which is identifiable with L(G) in the above way. Specifically our results are as follows, If His an abstract group with an invariant (in the sense of Section I) %-ring of subsets S and if there exist a representation of H and a&-homomorphism 77 of § which satisfy certain assumptions, s supports a unique translation-invariant O-finite measure » such that the conclusion of Loomis's theorem holds with L°(H, S, .) substituted for L(G). Further, we have a result which adds to the interest of the preceding. If to our hypothesis on H and s we add the requirement that for every a in H there exists an Ein s ~39- such that WE - (aE)) } 0 then H can be given a topology of the sort that a) His a dense subgroup of a locally compact group, b) the re- presentation is a strongly ‘continuous function on H, and c) the neigh- borhood system defining the topology is equivalent to a neighborhood system composed of members of Ss which have positive, finite p-mea- sure. Loomis's theorem is obtained from our results by showing that wo) is identical with the family of Baire sets having zero Haar mea- sure. This fact, combined with Loomis's assumption that the given representation of G is strongly continuous, then yields our assumptions on the representation and the 7~homomorphism as simple consequences. We begin the discussion by making a number of definitions. Definition 1: An Abelian ring of projections A on a hilbert space A is a family of projection operators on R which is commutative with re- spect to operator multiplication and has the following additional proper- ties: a) If P, Q belong to A then PQ and P - PQ belong to A. rr) b) Given any sequence {p.3 i =1 of elements of A such that iv i= n yee P,P; = Ofori fi there exists a member P of A such that4{ > Pit i=l n=/ oo converges weakly to P, i.e., > (P,, €,n) = (PE, n) for all € and nina. i=l Definition 2: Consider a non-zero element B of an Abelian ring of projections ‘A and any collection fad xel of elements of A such that A B= Ay and, if wz 8 . A,=0. Ay Ag If for every such collection at most a countable number of A, 's are non- -40- zero, then B is said to be countably decomposable relative to A. If all elements of A are countably decomposable relative to A then A is said to be countably decomposable. A system which, as we shall see later, is a very appropriate example of a countably decomposable Abelian ring of projections is the following. (As can be perceived from (5) this example is in most respects typical of the general Abelian ring.) This is that subset of the L™ of a measure space which consists of projections of the L? of that measure space, namely, the collection of all members of L™ corresponding to characteristic functions of measurable sets. Definition 3: An Abelian ring of projections ona hilbert space Rk is said to be full if no non-zero vector is mapped into 0 by all members of the ring. As we see from the fact that the set on which a measurable func- tion (if not equal to 0 n.e.) is non-zero has positive measure, our example of a countably decomposable Abelian ring also provides us with an example of a full Abelian ring. Definition 4: The union of a family ed os of elements of an Abelian ring of projections Aisa projection P such that 1) P, P= BP, for allie, 2) If for any QinA P,Q =P, for allxthen PQ =P, The union off Sy shall be denoted by LU Pl. It is a consequence of the second defining property of @ union that it is unique. | Lemma 1: Let A be an Abelian ring of projections on a hilbert re) space h. If fp} =1 has the property that PiP; = 0 if i fi, then -41- ee) , a) . 00 UJ P, exists and f= P. i| converges weakly to LJ P.. i= 1 i=l n= 00 i=1 Proof: Suppose we have P.} such that PP; =Oifi? j. _ i=l We know from the definition of an Abelian ring of projections that there exists PinA with the property that [s= P| ° converges weakly to P. Thus, to prove this lemma, we shall simaply verify that P has the two properties defining U P., since they define it unique- ly. a Let € and n be any two vectors in A . Now we see first that for alli (P;P€, n) = (PE, Pn) lim (( = P,)E, Pin) n-> 00 j= =m (( =~ P,P ,)é, n) n—->0o j=1 which from the arbitrariness of € and 7 proves P,P = P,. Next if Q in A has the property that P.Q = P, for alli then (PQE, n) = (PE, Qn) = lim (( 5 PE, 28) no i=l =lim (( s- P.Q)E, n) n-—*0o i=l . n =lim (( D_ P.)é, 1) n—-> co i=l = (PE, n) , so that PQ = P from the arbitrariness of & and Le So we see that P has the defining properties of the unique element iw P, and that, as a re- izl sult, x P, converges weakly to U | P.. isl -42- (The next lemma is essentially Lemma 2.5 of Part II of (5), with appropriate modifications.) Definition 5: A separating vector for an Abelian ring of projec- tions ona hilbert space R is a vector € such that if for any P inA Pé = 0O'then P = @, | | Lemma 2: Let A bea countably decomposable Abelian ring of projections with identity on a hilbert space. Then there is a separating vector for Ronk. | Proof: Let F = ips x be a family of non-zero projections in A maximal with respect to the following properties: 1) the members of F are mutually orthogonal, 2) PA has a separating vector for every P, in F. The existence of such a collection follows on applica- tion of the Maximal Principle. We see that because of the countable decomposability of the identity projection in A fps x must be countable.or is the empty set. In order to construct a separating vector for A we first prove that () Px exists and “ U p, =1. KeK The existence of LJ Fy follows from the countability of fp 3 «eK mae and from Lemma 1. The remainder is obtained thus. Letting o be an arbitrary non-zero vector in (I - C) Px ) - KeK not annihilated by all members of (I - C) Py JA, we show the existence — KeK in (I - C/) Px )A of a maximal projection annihilating c. To this end, KeK we first apply the Maximal Principle to show the existence of fod. a. a family of non-zero projections in (I - C/ Px yA maximal with respect aecKk to the succeeding properties: 1) the members are mutually orthogonal, -43 - 2) Qo = Oforevery6inL. Then we note that : UJ Qs exists and _eL _ annihilates .y by the countable decomposibility of A and Lemma 1. Finally, aif Sin (I - U Py \A annihilates C. then S( LJ Qs )} = 5S, for, weK 6eL inasmuch as [sq - Ue Pp, - U J a4)5IF = ((I - os P. -U a,)5, 8) = S(I - U PP. - (S} Qs) must vanish due to the maximality properties xe K 6eL of fos 5eL* Since (I - UJ P SF 0 by the choice of 0, then =[- C) Py “ey Qs is a non-zero projection in A. Ris orthogo- we K 6eL nal to L) Py, , and so by Lemma 1 orthogonal to every Pe « Further, xeK Qs being maximal among those elements of (I - P )A which beK eK annihilate $ , RA has C as a separating vector. But then ip,3 irs properly contains fp, J K and has the same properties, a contradiction of the maximality of fo eK’ Hence, I - (J P, is “ “eK the zero-projection, or, U p «1, KeK We can now construct a separating vector for A. From the defi- nition of fp} eK and the countability of K, there exists a countable collection of vectors de iy such that 1) GK is a separating vector eK for Py ‘A for allw, 2) Py = £2 and 3) € = 2_ ox exists as a limit ove K in the norm. Then, if we recall that 1S, P =tJIand if we remember aKeK 0 that by Lemma 1 tor any countable disjoint collection fos! =1 of projections in Af. > ie " converges weakly to UO Q., then € =1 i=l is + actually a vector such as we desire. For if P€ = 0, we can conclude that: - -44- >_ ker, eM =D _ lew, e[[? xeK “xeK d_ (P, £, Pé) weK (é, PE) IIpell = 0 , so that PP, = 0 forall&. But this means that for every n in k JPnl* = (1, Pn) d_ (Py n, Pn) “Ke K a ((PP, )n, 1) xKeK 0, iee., that P=0. Thus, by definition, € is a separating vector for A. Lemma 3: A full, countably-decomposable Abelian ring of pro- “jections A of a hilbert space Rk contains a collection of non-zero pro- jections ip, i such that 1) P,P, = 0 if x # 8 and 2) given any P in el B A wi P = (PP, ) iat OM for some sequence of a's. Proof: We can prove this lemma by taking EPS ey to be any maxi- mal collection of mutually disjoint projections in A. (This exists by the Maximal Principle.) Then, since for all 8 in I and every non-zero Q in A -45- P, [e-U ae,)] = P, | -a U cap, | «el Kel QP, f 0 QP, #0 = P,Q - ex U (QP, 1 QR, f 0 = PQ - P_Q = 0, is contradicted if Q - () (QP,.) = 0. wel QP, la 0 the maximality of fed The collection of «'s for which QP, f 0 is countable in virtue of our assumption of countable decomposability for A. The indicated unions thus all exist if we invoke Lemma l. Definition 6: A @-homomorphism of a O-ring S of subsets of an abstract set onto an Abelian ring of projections Aisa mapping 77 of § onto A such that | 1) 7 (ANB) =77 (A) 7 (B) 2) (AUB) = (r(A) - 77(A)m(B)) + TB) 00 3) If fat is a collection of mutually:disjoint ele- _ n* i=l oo oe 00 ments of S then é> 7 (A,) converges weakly tom( (/ A,). i=l n-=l i=l Definition 7: A function f on a measure space to a hilbert space k is called a countably-valued step-function if there is a countable, 00 mutually disjoint collection of measurable sets fzf and a se- Le i Pe) i=l quence of vectors in 5 §; such that for every x i=l 90 (x)= S 6: Le (). i=l i =-46- Definition 8; Let f be a function on a measurable space (M, 5) to a hilbert space Kr and let 7 be a 9 -homomorphism on S to a family of projections on k. Then f is said to be strongly measurable relative i? @] to 7 if given any member E of S one can find a sequence fe J n=l of countably-valued step-functions with the following special property: oO of measurable sub- 00 there exists an increasing sequence fz 3 n=1 oo = sets of E such that 7(E - ( J E,) = 0 and such that on each {iJ n=l converges uniformly to f. nel As an example of the above sort of function let us consider ¢(x)€, where € is an arbitrary element of L(G, B, .) and where ¢ is the "left regular representation" of G, i.e., the representation in which each element of G is mapped into the corresponding left-translation opera- tor on L(G, B, LL). (B is the 7-ring of Baire sets of G and y is the Haar measure on 8.) Our measurable space is (G, B) and ff is any 7 -homomorphism of B onto an Abelian ring of projections on — L(G, B, 2}. As shown on p. 41 of (8) ¢{x)& is continuous at the identity. Clearly ¢(x)€ is uniformly continuous in the following sense: if N is a neighborhood of the identity e such that to (aye ~ d(eyell fe for xin N then for y = ax in aN llstyve - a(arell = Hotere - stare Ils(ard(a)e - d(a)d(eyell Ild(arl -lIscere - (ere Ilsecre - a(e6 ll <e. " Also, B being generated by a family of compact sets, every member of B is contained in the union of a countable collection of compact sets, so -47- that any E in B is contained in the union of a countable number of translates of any open member of B. Adding to this the fact that any the identity contains an open Baire set which in turn ) contains the identity, the "uniform continuity" of ¢(x)€ makes ita neighborhood of uniform limit of countably-valued step functions on E, and so, a fortiori, ¢(x)€ is a strongly measurable relative to 7. Definition 9: By a projection-measurable group we shall mean the ordered quintuple composed of: 1) An abstract group. G, 2) A@-ring 5 of subsets of G which is invariant in the sense of Definition 5 of Section I, 3) A hilbert space k, 4) A @-homomorphism 7rof S onto a full, countably de- composable Abelian ring of projections on k, 5) A representation ¢ on G to a group of unitary opera- tors on k such that: a) ¢(x)€ is strongly measurable relative to 7 for every €in R, b) For every x in Gand E ins N(xE) = d(x!) a(E)d (x). We shall denote a projection-measurable group by the symbol (G, 435, 73h). Proposition 1: Let (G, 3:5, 1:k) be a projection-measurable group. Then the following assertions are true. First, there exists on Sa left-invariant, @-finite measure unique up to a multiplicative constant, with the property that it vanishes on precisely. those sets of S which belong to roo). -48- Second, G can be given a topology T in which a family of neighbor- hoods of the arbitrary element x is given by the family xU of all sets “of the form: ; x Eylu (yz - E) ULE - yE)) <e, 0<e 208) <eoh all such sets have positive measure and, when x is the identity, are symmetric. The subcollection xN, consisting of those sets for which wes) is finite, has the properties that it is a family of neighborhoods equivalent to the defining family xU, and that each member has finite measure. Special properties of T are: a) for any neighborhood U of the identity there exists a neighborhood V of the identity such that vv-'cu; b) given any neighborhood V of the identity and any a in G there is a neighborhood W such that aWa ‘CV; c) there exists a neighborhood of the identity coverable by a finite number of left translates of every other neighborhood of the identity. Proof: We obtain our first conclusion as follows: 1) we con- struct on Sa @-finite measure m such that left-translation is a non- singular transformation and such that m(E) = 0 if and only if NW(E) = 0; 2) we invoke Theorems 1 and 2 of Section I to assert the existence and uniqueness (up to a multiplicative constant) of an equivalent left-invari- ant @-finite measure. Let fel be a collection of mutually disjoint elements of r(S) such that for any other element Q of 1 (S) -49- oo a=U @P). isl i ' (Such a collection exists by Lemma 3.) Then if for each & in I we choose for Pw (S) a separating vector ew (existing by Lemma 2), we can define a real-valued function m on Sas follows: for every Ein s ZH (WEN Es Ge) TUE) f 0 m(E) = T(E)Rf 0 0 , 7(E) = 0. Since if mE) f 0 then W({E)P, 7? 0 for some & in I, we can see that m(E) = 0 if and only if 7(E) = 0. That left-translation carries a set on which m is zero onto a- nother set on which mis zero is seen by combining two facts. The first is that we have assumed for all a in G and E in S the formula: r(aE) = dla” (E)sla), from which we see 7 (aE) = 0 if and only if 7(E) = 0. The second fact, established in the preceding paragraph, is that m(E) = 0 if and only . if w(E) = 0. Next we show? -finiteness. In view of the properties of the collection inj we have for any E in S and a suitable sequence of i?) A's, fa.f i=1? 1 )UMEIP, MEE, ) = 0 ifi, ae!’ 2)W(E)P, #0, a 8 a)mE)= J wep, . i=l i - a) We choose a sequence of sets in S, fz, t=1? such that -50- EME, =, id, “and (1(E,) =P ast Then, keeping in mind the definition of 7 and the fact that Lemma 1 n 00 ee) makes $5 reNz,)} converge weakly to LJ T(ENE.), n=l i i=l i=l 1 oO T(E) = O T(E)P, is oO U T(EN E,) i=l oo m( LU) ENE,) i=l = meym (J) =) i=l 8 whence, 00 00 me (ENC 8,))= 78) - mE)W( OB) L1= i= : = 0, ‘or r* 3) m(E - (EN) £,))=0. i=l Further, for every i mE) = 2 (r(B,)P, bs 6) w(E.)P, f 0 2. (Py Fae bac» Bx) axel P 0 Po, xf (PP é " »€ ). i ~51-- Hence, ‘since E = (E ¢ Cy E. NUE - U E. i)s E is the union of a i=l |: i=l countable collection of sets for which m is finite. Finally, we prove the countable additivity of m. ' Consider any ce) countable, disjoint collection of sets, lz,J . In the case that i=l m(E. ) = 0 for all i, we know that T(E, )=0 for alli, and this, combined i) with the property of 7 requiring that is T(E. ay converge o i=l n=l weakly to 1( (") E. 4) shows T( Y E. i) = 0 and shows countable additi- i=l vity. In the remaining case where en(E,) 7 0 for at least one i, let fo} -luetlTia m(B,)P, 4 04, J j=l 1 and observe that co >= m(E,) = Ss (T(E; )Pee 58.5 Ex 5) i=l *. a oes = s pa (TE, )Pac S052 805) y= i= sm U E.) Py Eas Eq :) j=l izl 1 jos J Cs and, since I( E.) Py 7 0 if and only if T(E, ) Pox # Ofor some , i ye i} rr i 6] TL) EiIPx jf 5 £4) =U BR by, 6) eel i=l Laws " re ioe] m LJ E,)P, #0 i=l 00 = m( (JE i=l Thus, we have proved the existence of a left non-singular, 7-finite measure on S for which m(E) = 0 if and only if T(E) = 0. Theorems 1 and 2 of Section 1 now assert the existence and uniqueness of an equiva- lent, left-invariant, O@-finite measure. -52- We have now shown, by proving the preceding, that the hypo- theses permitting the first conclusion of Theorem 3 of Section lare satisfied; hence, the second conclusion of the present proposition, which is precisely the same as the above must hold. - Definition 10: A set E in a projection-measurable group (G, $35, 7 ;k)-shall be called measurable if and only if E belongs to S. By the measure of a measurable set E in S we shall mean w(E). Definition ll: By N we shall mean the family of neighborhoods N at the identity of Proposition 1. Definition 12: We shall say that a function on G is continuous if and only if it is continuous in the topology T. Definition 13: We shall say that a subset of Gis. a neighborhood of a point x in Gif it is a neighborhood of the point x in the topology T. Remark: Of the following sequence of results, Propositions 2, 3, and 4, as wellas Theorem 1, constitute what is essentially a repro- duction of Loomis' main arguments in (3), while Lemmas 10, ll, and 12 ‘are known in the literature; the others are original and arise from the necessity of using the not quite locally compact topology T to prove results obtained by Loomis from local compactness. Lemma 4: Given a projection-measurable group (G, 4;5,7 sh), _then for every & ink ¢(x)€ is a continuous function on G toh. Proof: Proposition 1 of this section states that an arbitrary mem- ber of N has the form fx\u(xE - E)U(E - xE)<e, 0 <e ¢2pn(E)< wos . _ Accordingly, it will suffice to show that for every n> 0 and every element € of R there exists a set F in § such that -53- MF) 20 and Ilscoe - ell<n for every x in FF’, Let n}0 and teh be given. Then using the definition of strong measurability with respect to 7 we see that there is a K in § for which | w(K)> 0 and that there is a J in A for which [lye - Slen/4, when x is in K. Then for some fixed x, in K we have sce - ot, )6H ellos - sll + Hoc,re - 5H <n/2 for every x in K. Or, since d(x,) is unitary, Iseoe - ell<are for every x in F = x, K As a result, if rand s are any two elements _ of ar Ils(rs" ye - elle Worse - d(xyell + Ione - ell = Idlers "VE - d(s)éll + Imre - ell = foisyé - el + Mere - ell <n. | But this is the same as saying that for every x in Fro! IIdxé - éll<n. Lemma 5; Given a projection-measurable group (G, 4:5,7 ;h), there exists a measurable set E having positive, finite measure, a non- zero vector Eo in kh. and a positive constant k with the following proper- -54- ties: 1) Eis coverable by a finite number of right translates of each neighborhood of the identity, '2) For every measurable subset A of E { = (W(A)E,, &,) <kH(A). Proof: By Proposition 1 we can choose an F from the neighbor- hood base N such that p(F) is positive and finite and F is coverable by a finite number of left translates of every member of N. By Pro- position 1 the fact that p(F) > 0 means that mor) / 0, ‘so that, 1 (8) being countably decomposable, then on TEM ME 78) = T(FA 8)) must bea countably decomposable Abelian ring of projections with identity. It follows by Lemma 2 that w(F NS) possesses a separating vector in T(E)K, a vector we shall call Eo" Hence, the function n on FN § de- fined for every EeF Nh Ss by the equation n(E) = (T(E)E,, £4) is a measure on F /) s, and n is equivalent to p on Ff) s. The proof of our last assertion is a result of the following two remarks, For fet" » an arbitrary disjoint sequence of members of S, the definition of 7 requires that 7 ( © E,) be the weak limit of io TtE,)3 ° C Further, from the definition of n and the fact that é. is a separating vector for 7 (Ff) 8) we have that n(E) = 0 if and only if 1 (E) = 0, while we know from the first conclusion of Proposition 1 that p(E) = 0 if and only if 77 (E) = 0. Now by the Radon-Nikodym Theorem there exists a non-negative function f integrable on the measure space (F, F\S, ») such that n(E) = f(x) du(x) E ~55- for every EeF f\ Ss. Then, inasmuch as . n(F) = (™(F)E., &), we know that 0 nF - (J fxerls(x)én}) = 0. n=l It follows that for some sufficiently large ny E = {xeFle(x) én, has positive, finite p-measure, So, we see thatfor every KinE § (W(K)E,, &,) = n(K) = / f(x) d(x) K Z é ie du(x) n, vA Hac) dysto) nH(K). This is the second conclusion. To obtain the first conclusion, let us consider E, F, and V, an arbitrary member of N. By the construction of E and the choice of F ECF and . n FC { J bv. izl Since by Proposition 1 F and V, as members of N, must be symmetric, we have E on Va, , i=l which gives us the last conclusion of this lemma. Lemma 6: Given a projection-measurable group (G, $;5,17 ik ), the product of any finite collection of measurable sets open in the topology T -56- is itself measurable, Proof: Let R and S both be measurable and open. Then, inas- ‘much as Proposition 1 shows that every member of N is symmetric and that for every x Nx is a base of neighborhoods at x for the topo- logy T, sg is open as well as measurable; thus, rs =dxks"'N Rr = 93 = Eluces'N Ryo . But the latter set is measurable by Corollary 1 to Proposition 2 of Section I; whence,. RS is measurable, Having proved the present lemma for the case of two open, mea- surable sets, we see that the method used is applicable to the case of n+1sets, narbitrary, if the lemma is assumed in the case of n sets. But this means that we have found an induction proof of our lemma. Lemma 7: Let (G, 4:S,7 3k) be a projection-measurable group. Then for any N in N and any ¢€70 there is an M in N and a measurable set K, open in the topology T, such that KCKMM'CN “and M(KMM"! ~ K)<en(KMM"'), Prdof: Given N in N, then, as asserted by Proposition 1, there is a W in N for which ww en and wewl, As a result, we have that there is W in N uch that w-CNn. -57- , z By induction we can construct the sequence in,J of mem- i=l _bers of N such that for all i: 1) N, belongs to Ny (= ineNINCWwH ), 2) 3 Ni +1 CN; . The proof is based on the following assertions of Proposition 1: 1) for every N in N there is an M in N such that MM en, 2) N= No! for every N in N. We can see that for alli Lim WN. CWN, j-> 00 +1 =(WN, , ,)N., 2 i¢ Vv ig¢i Cwn, . And since for alli WN,CW°CN and since Lemmal6 holds, the following sequence of inequalities re- sults: ZL = lim L(WN;) j—p 00 £u(WN. |.) i¢l £u((WN. | |)N, Lt itl *) £u(WN;). Thus, there is an i for which Z guCWHN; utWN, N, ct +h. 2) And so,. 2 + Lowa, IN, 2 - ws, Jet, ) -58- ~ Now, let K = WN, ° and © M=N. , i re) and let us remember, on the one hand, that WN, is open and measur- re) able by Lemma 6 and, on the other hand, that N,; =N, “I Then we fo) fe) see that we have found an open, measurable set K anda set Min N such that —CKMM™ and | m(KMM 7! - K) <eyu(KMM7|) Since, further, Lemma 6 makes it clear that KMM7! is measur- able, and since KMM”! = (WN, )N, (x, )7! ’ (oun ©) re) = (WN. )N, * 1 1 - oO fe) C wn, é, - 9 CWwWw CN, our proof is complete. Definition 14: Let kK be a hilbert space, (M, S, m) be a measure space, and f a function on M to Avsuch that (n, {(x)) is integrable for every nin k, Suppose that g(n) = I (n, £(x)) dm(x) is a continuous linear functional onf. This means (see p. a of (4)) that there is a unique Cin R such that -59 We may thus define the integral of f(x) over M Sete) dm(x)) by the J tx dm(x) = q. M _ By the very definition of J ite) dm(x) we have the following for- M ' equation mula holding for every ny in k : (n, Jie dm(x)) = J. tnvscen dm (x). M Lemma 8: The following are properties of the integral defined in Definition 14: 1) For every integrable f on M to (n, I £(x) dm(x)) = I (n, £(x)) dm(x), 2) The family of integrable functions lis a linear space over the field of scalars belonging to kK and the integral is a linear operation on I, 3) If T is a bounded linear operator on K and f(x) is inte- grable then T(f(x)) is integrable and . T(£(x)) dm(x) = T vA £(x) dm(x) , M M 4) I co N M= UA, , i=l where the A,'s are measurable and 1 fi ANA, = ¢, and if f is inte- grable on M and on all A:, then we have for all nin iN oO 5, (n, [ f(x) dm(x))= D- (n, iA f(x) dm(x)). i i=l Proof; The first property is immediate from our definition of in- tegral (Definition 10), while property 4 is the result of combining the first property with the fact that the indefinite integral of a real-valued -60- integrable function (integrable in the standard sense) is a countably additive ‘set function. - We can now dispose of properties 2 and 3 by comparing our defi- nition of integral with Pettis' definition for the same situation. Because of the standard representation for continuous linear functionals on a hilbert space (mentioned in Definition 10) property 1 of our integral im- plies that the two definitions coincide. But properties 2 and 3 are pro- perties of the Pettis definition, and, hence, of our definition. (See (2). Here the underlying measure space is assumed to be totally @-finite, but this does not affect the properties under consideration which only require those properties of the abstract Lebesgue integral that are true in general.) ( Lemma 9: Let R be a set of finite measure contained in a neigh- borhood N of the identity which ig coverable by a finite number of left- translates of any other neighborhood, Then there exists S608 dm(p) for every € in kh - HER= CO R,, where the R, are measurable, then for =1 alli (d(p)E) dm(p) exists and . R, n , Ls1008) dm(p)=D— / (d{p)&) dmi(p) . R i=l R; Proof: Since by definition of G(p) d(p)E is strongly measurable relative to WT on R for every &in k , then by Lemma 4 ¢(p)€ is continu- ous. Also, in view of Proposition 1, an N of the sort hypothesized actually oxi sts. Thus, ¢(p)€ is the uniform limit of countable-valued step-functions on R, and (n, 6(p)€) is measurable on R. Combining this with the fact that . -61- lin, H(oé)] <lhnll fldter€ll = Hnll- lle, it follows nat / (n, d(p)E) dm(p) exists for all n and & ink and defines for each fixed € a continuous linear functional on h. Hence, the re- quirements of Definition 10 are satisfied so mat /idlo)é) dm(p) must exist for every € in k . n If R= > R., where the R,'s are mutually disjoint, then pre- i=l cisely the argument given above to show the existence ot / (dlo)8) dm(p) R will suffice to show the existence of / (¢(p)&) dm(p) for alli. Now we R i see from property 3 of the integral defined in Definition 10 that n Jor dm(p) = S$ J ((p)) dm(p) R i=l R, Proposition 2: In a projection-measurable group, {G, 43S, mk); there exist R,D, and V,three measurable sets of positive, finite measure, with these properties: 1) RD cv, 2) There is a vector € ink such that if n= oy L d(p)7(D)E dylp) > then TR) # 0. Proof: We know by Lemma 5 that there exists a measurable set F of positive, finite measure, a positive constant k, and a non-zero vector € in T(F)A such that 1) F is coverable by a finite number of right translates of each neighborhood of the identity, 2) (7 (E)E, €) < ky(E) for every measurable ECF. We can assume, in addition, that 2 él = w). And, since we know by Lemma 4 that ¢(x)€ is continuous, we can choose by Proposition 1 a neighborhood W of the identity such that We woh W has positive, finite measure, and for every x in W Ild(xré - ell< $ Hell . Then by Lemma 7 we can find a measurable, open set K and a member M of N, both having positive measure, such that K CKMM™! cw, KMM7! is measurable, and i M(KMM™! - K)<(—e )(KMM™!), Now suppose we define the vector 5 by the equation 6=——, f _, Hx) dm(x). u(KMM™') ” KMM (The integral exists by Lemma 9, by the fact that KMM cw, and by the inequality | w(KMM™!) u(W)<eo.) Then, observe that € is integrable over KMM~! and remember that our integral is a linear operation; we have, if Ils - é\| is positive, 6 - ef” = [6 - €, i bkp)é dilp) - af dus(p) lb - Ell° = L Tone eet (p)€ du(p) aM) 16 Hp) w(KMM~") KMM [s-& —_j A _(6(e)é - &) dulo) ] w(KMM~) ~KMM art an (5 - €, 4(p)& - €) dulp) b. -63- <1 f Hs - sll (3 fl 6lly ate) uw(KMM~') ~KMM = jo - efi LEH so that in every case — IIs - ell<} Hell. Next, expanding (6 - €,6 - €) shows that | Re (5,6) => ( [lll * + Hell? - Ils - ell*. Hence, combining this with the fact that Ils - ll<s Héll, ((6, €)]> Ref (6, €)] = ( (ell? + (ell? - [fs - €l]?) ds (Hell? + Hell? - Zilell*) ry (llell® + Fill? - zilell*) 1 2 1 = > Ilell = > HF). Now since F was chosen so as to be coverable by the union of a finite number of right translates of every neighborhood of the identity, n rc Wa i=l i As a result, n Fo O C,, where the C,'s are measurable, mutually disjoint, and for alli Cc. ,;CWa,. Further, by Lemma 9, ce = ae 1 9p mC, )E dyu(p) exists -64- for all i, so that by the linearity property of the integral of Definition 14 and the fact that m(F)& = € 62 & C.. i=l Thus, as we know. (6,6 )1> 5 HF) it must be that for some ig 2, . elds wc, ). oO Le) 1 So, setting i C. , C= GC; > a =a, wecan say: 1) fe) re) ) a] > snc), where v= 1 / _ HedTIC)E ap(p) y(KMM)/KMM and CCF) Ma; 2) L=KMa is a well-defined measurable set, where the measurability of L comes from Lemma 6, But this last assertion proves the present proposition, if, also, T(L)S #0 and Lc7! is a subset of KMM™!. | We can show from the preceding that m(L)S # 0. Suppose mL) 5 = 0. Then, using properties 1) and 2) of the integral of Definition 14, the commutation relative between ¢(p) and7 (E), and the identity. ME)E = &, -65- (6,5) = (€, 9 - Lys) -=(¢,—l— L blo) (CIE aul) - +9) d(py(C)E du(p)) w(KMM {y¥KMM"! prmens O aeM}) KM! , me (¢, — _/ _ Ble) m (CE - 7 (L)d(p)7(C)é] dup) w(KMM~ )“KMM (€, [4(p) r(C) - T(L)d(p)7(C)} &) duc acne! [sted preted 8) duce | tate (m(eCN FE, T(pCN L°)d(p)7{C)E) dulp) iu (Here L® is the complement of L.) But L°f\pC is void unless peLoC™! = (KMa)°c™! Cc (KMa)°a~ ‘m7! = (KM)°M"! CK , _ so that the last integral need only to be taken over KMM7! - K. Further we have Ince A pcyelle Vawter Apc) SVeu(pc) — =Feut) and Ilr cy sermorellalmoyell < Heute), where we use the special properties of F and € as well as the left in- variance of 3; this shows that (nce Npcyé, TILEN pcysie) icielé kulc) . -66- Hence, we see that 3. lel tf [ror pcre, LEN pcrdipyricyel awe) w(KMM~)*KMM~ -K ua) uM! (ku(C)) du(p) pL . -K ; -1 QKMM -K)) (ey(C)) , w(KMM" |) and, remembering the choice of K and M, -1 K - it ( EMM HB) ) eu(c) <( Se) H(C)) u(KMM ) = . H(C). But since we have shown already that le el>duen, we see that assuming w(L)S = 0 leads to a contradiction. If we combine this result with the observation that Lc! = (KMa)c7! C(KMa)(Ma)™ =KMM~! , we see that L, GC, and KMM7! are sets of the sort proved to exist by this proposition. | Proposition 3: Let R, D, and V be the three sets asserted to exist in the preceding proposition. Then if E is a measurable set such that EC (aR) [) (bR) for some pair of elements, aandb, of G, andif -67- Ll n= stn |, somo diu(p), then — ME)d(a)n = WE) d(b)n. Proof: Since by our hypotheses that ECaRfMbR and RD Cv we have ED™'C (av) () (bv), it results that vA T(E N pD) dlp) du(p) = Lm MpD)d(p)& dpl(p), aV T(E (\pD) being 0 except when p belongs to ED”, Thus for all n ink, (n, J we NpD)d(p)& du(p) = (n, [me ApD)d(p)E du(p)), aV bV or | It W(E pD)d(p)é) dulp) = [ (n, T(E pD)d(p)&) dup). aV bV Then, by the left-invariance of p, A (n, T(E NapD)d(ap)&) du(p) = J IE (\bpD)d(bp)é) du(p), so that [vee NapD)dlap)é dp(p) -[ (EN bpD)d(bp)E dp(p). Next, since the definition of a projection-measurable group requires the identity d(x) W(S) = 17(xS)¢(x), -68- L weestorrmor dis(p) - | ne2iseprrtoi du(p), ‘Finally, we know that ¢ is a representation of G, and we know, by conclusion 2 of Lemma 8 that bounded linear operators commute with the operation of integration; hence, reve) | snyne di(p) = reewon / spree du(p) , or, by the definition of n, W(E)d(a)n = W(E)d(b)n. Lemma 10: Let E and F be measurable subsets of G, both of positive measure. Then there is an at most countable disjoint col- lection of mutually disjoint, measurable subsets of F, fad > such that “et A,Ca, E for every Xin I and we - Ur a,yso. Proof: Since pis a O-finite measure, E and F are each the union of an at most countable disjoint collection of sets of positive, finite measure. Thus we need only prove the lemma in the case where F is of finite measure. Consider the family 2 of all collections of disjoint measurable subsets of F such that each subset is contained in a left-translate of E. This family is non-empty since we know by Corollary 2 to Pro- position 2 of Section 1 that p(xE NF)? 0 for some x inG. ‘Let tyes be a subfamily of the above totally ordered with re- -69- , spect to the ordering relation of inclusion. If we define F to be the union of all sets occurring in at least one of the Fy 's we see that F is ' an upper bound for fey yes" Thus, we may apply the Maximal Prin- ciple to conclude that g contains an element H which is maximal with respect to the inclusion ordering. We shall conclude our proof by showing that H is at most counta- ble and that, if H = Hed er ; we -Un axel x ) = 0. The fact that H is at most countable comes from the fact that itis a collection of mutually disjoint sets of positive measure all contained in a set of finite measure, for the latter type of collection is known to be at most countable. And, since we can now say that F - U Hy, is mea- xel surable, the second property of H follows: if W(F - (J H,)> 0 Xel then for some x u [xe No -U H, J >0 “el [xE Nie - U Hy Ui) vel and is a member of Ce properly containing fu} wel =H contradicting the maximality of H. - Definition 13: If (G,¢; 7, 5:4) is a projection-measurable group, let real (complex L(G, Ss, wt) be the real (complex hilbert space corres- ponding to the family of all real (complex)-valued functions square-inte- grable on (G, s, 4). In the following, the symbol. “L2G, S, »)'' shall be prefixed by the adjective 'real'' or the adjective ''complex" only if the -70- context’ does not make it clear which is appropriate. Definition 14: Let (G, Ss, LB) bea left-invariant GMG. If f is the element of L(G, s, .) corresponding to a square-integrable f, then for every ainG and every E in S we can define the operators T, and Pr by the relations: 1) T,() = fo where | f(x) = f(a x) Nnee., 2) Pi (f) = fp» where f(x) = PreglbE lec). ne, Since wis a left-invariant measure, qT. and Pp are both bounded opera~ tors on L(G, 5, p). Lemma ll: The family of all real (complex) linear combinations. of characteristic functions of sets of positive, finite measure is dense in L(G, 5, p). | Proof: (See Exercise | of section 42 of (1).) Proposition 4: Given a projection-measurable group (G, é;8,7 ;K), there exists R, a subfamily of S, with the following properties: 1) forall AinS ANRC: R; 2) forallainG aR CR 3) the collection of all real (complex) linear combinations of characteristic functions of members of R is a dense subset of the | real (complex) L7(G, S, 2); -71« ) 4) R can be mapped. into kK by. the function d in such a man- ner that for all E in R a) If pis an arbitrary member of G MoE) = dp [hz] , b) If A is an arbitrary member of S Man =) = ma) [he], c) Wheel)? = woe). Proof: Let R the family of all measurable subsets of left trans- lates of the measurable set R of Proposition 2 of this section. We pro- ceed to verify that it has the four asserted properties, Property 1) of Risa consequence of the fact that R. is a sub- family of 5 and the fact that S is closed under the operation of intersec- tion, Property 2) follows from observing that if ECbR then aE C(ab}R ‘and that translation is a measurability-preserving transformation ona generalized measurable group. Property 3) results from combining the preceding two. lemmas. For Lemma 11 shows linear combinations of characteristic functions of sets of positive, finite measure to form a dense subset of L(G, Ss, Ht). And we observe that Lemma 10 shows any measurable set in G to be, up to a set of measure zero, a disjoint union of sets in R. Hence, combining the preceding remarks, we see that any f in L(G, S, .) can be approximated arbitrarily well in the L*-norm by a linear combinations _72- of characteristic functions of subsets of R. Proof that R possesses the fourth property asserted will be obtained by first defining a function Aon R to h and then showing it to have the desired characteristics . We define \ for an arbitrary member E of R. by the equation, ME) = T(E)d(a)n, where ais any element of G such that ECaR. and 1 is the element of kh defined and discussed in Propositions 2 and 3 above. We observe first that by Proposition 3 dis uniquely defined for every E inR. Further, since by Proposition 2 T(R)q f 0 and u(R)>0, n can be assumed to be such that llr¢Rynll® = wR): thus, by definition of A 2 [TAcryll * = pcr). We see that » has the first two desired properties by some com- _ putations. For, in view of the definition of \ and the commutation re- lation Bp) W(E) = T(pE)¢(p), we have d(p) [Mey] = 6(p) rte) dan] = 1(pE)d(p)d(a)y = T(pE)d(pa)n = ME), and ma) LAceyd = tay fre) darn] -73- T(ANE)dl(a)y MAN. E). That ]\ce))) * = wae) for every E in R will be shown as follows. We shall construct a left-invari- ant measure V on S such that . v(e) = fAceyll? for every E in R.. Since an invariant measure on S is unique up to a multiplicative constant by Theorem 1 of Section 1, since we know that for the set of positive measure R w(R) = | hiRyll 2 = V{R), then p and V are the same measure. Hence, for every E in R |ceyll * = 7 (e) = p(E). We construct the measure V/ using Lemma 10. By Lemma 10, any measurable set E has the form tre) E = PU (J z,] izl. ire) where »(P) = 0 (so that by Proposition 2 m(P) = OJand fe i=l is a mutually disjoint collection of subsets of translates of R.. Then, by definition of R,, d (E;) exists for all E,, and for an appropriate sequence o of elements of G, ia,3 i =1? ME,) = 1(E,)d(a,)n. We define V for E by setting ~74- -(B) = = dz, yl * First of all, V is uniquely defined. Let the arbitrary measura- ble E of the preceding paragraph have a second decomposition: E=Q uf O er. i). i=l Here, of course, Q and fr} =] have the same properties as, re- 00 spectively, P and io ie? and ME;) = T(E; )d(b;)n. So for all i [|e pil? = [rceote,rall? = Ik.C. (E, F(a, yal 2 J = | = (rf U NF) Mada, olan - > (WEAF, 9, )n, (a. )n) fz = Ince, AF ll *. s (TUF )4(b,)n, d(b,)n) so = ok, ng E,]o(p, Jn, d(b;)n) i=l u Hence, > Ace Il? = jel hee it MeiMs Fe nat it a pte It em (ME, NF, )9b;)n, 6(b,)n) Ihre) Ace Il? Ms Ms Ket iT} _ fa i} fa (ce, Ne pil? fl 8 rear ce a ul = 27 | Rae il 00 Thus one could have defined J/(E) to be > hel 2 without changing’ i=l the value of the former, which shows V to be a uniquely defined function -75- = on S. Now, to prove that V is invariant under left-translation, we must observe first that if co E=P of 0 z,] ; i=l where P has measure 0 and the E.'s are mutually disjoint subsets of R, then aE = (aP) U(L (eB), where the left invariance of » gives aP measure zero and the second property of R makes the ak, mutually disjoint members of R. Asa result, WaE}= 5 [Max i=l = |v, kell? i=l S fA? i=l u = V(E). Now only the countable additivity of V remains to be proved. Let ee) fF, be a collection of mutually disjoint sets in S, and suppose that _ k=l w ik, }? =] is a collection of mutually disjoint subsets of LJ Fy all ~ k=l belonging to R, such that i°] wo HL) Fy, - L) K,)=0. k=l i=l 00 Then for every k, ir K, 5 i=1 has the same properties relative to 00 ee) . F,. as has {x} i relative to ~ 1) Fy: But this enables us to say, k=l remembering the definition of Vand assuming aK.) = W(K, )d(a,)n, that -76- = vie) = =| JAA Kl? =z=liel = = Se [re ydex,)||? =li-=l = z = [Ir ,) TK, )o(a,)all * = a = (WE, MK, )d(a,)n, dla,)n) = 1a = x S (TIE, 1 K,)d(a,)a, d(a,)n) = = (7r( O PNK )dla,)n, ¢(a,)n) L= = x (7(K;)9(a,)n, 9(a;)n) i.8) - 2 [Ack yl}? = v( O F,). ul Thus Y must be countably additive. Having proved that V is a left translation invariant measure on ‘S, it follows in the manner already described above that || Aceyll* = pce) for every Ein M. This concludes the proof of the properties asserted for d and, hence, of those asserted for M, ending the proof of this proposition. Definition 15: By a unitary mapping of a hilbert space hs onto (into) a hilbert space kh, we shall mean a linear, homogeneous mapping T on h, onto (into) h, such that if € and n are any two vectors of ki then (6,7) = (T(E), T(n)). It is clear that he. and T(h,) are either both real or both complex. -77- A familiar mapping provides a simple example of the notion just defined. This is the well-known mapping of the complex Lif, 7}) onto the following complex hilbert sequence space, a SS Is,1? <0] ffs}. =1 | s, complex, a Is] K of. Proposition 5: Let (G, $38, 73K) be a projection-measurable group. Let V be a unitary mapping of L(G, S, 4) onto R , a closed hil- bert subspace of A, and let V have following properties: 1) for every p in Gand fin L“(G, 5, ») (v"'d(pyv)e = Tf 2) for every AinS and f in L(G, 5, ) (volr(ayv) = Pat. Then the set of all unitary mappings of L(G, 3, w.) which send it onto Re and have properties 1) and 2) has the form fx | xascalar, [xl = if. Proof: (In the following we shall use the symbol for the charac- teristic function of a set of finite measure as the symbol for the corres- ponding element of L(G, S,2)-) We can verify directly that if & is a scalar of absolute value l, then the mapping «V is unitary and has properties 1) and 2) of V. Thus, we can restrict ourselves to showing that for any unitary mapping W of L(G, S, ») onto R having properties |) and 2) the unitary operator ¥(=w-ty) on L(G, S, 2) has the form: Xa scalar. Ix! must certainly be 1, wo! being unitary. ' First, it is useful to observe two relations satisfied by Y. For -78- all ain G and all E in s we see from properties 1) and 2) satisfied by V and W that. tr -vuv-w'uw, a a a whence -1 -1 VT_V =WT_W , a a or T Y=YT_. a a Similarly, PLY = YPp . Using these relations, we can obtain our result once we know that for every E of positive, finite measure (fe) = he), 6B being a scalar possibly depending on E. For it follows first the 8 is independent of E, since the contrary would contradict the fact that for any two sets of positive, finite measure, A and B, x fy) x Palhy UB) PAE, yg) Pal Yhe Up) = K, y As Up) is and, by similar reasoning, ¥(fg) =f Bs Then, inasmuch as " Y = «I on all characteristic functions of sets of positive finite measure, we see -79- that the set on which this equality holds can be extended to the set all linear combinations of such functions. But by Lemma ll the latter set is dense in L7(G, Ss, ), which, combined with the continuity of Y and L means that | Y=«ali on all of L7(G,3, .). Thus we have proved our result knowing that corresponding to every E of positive, finite measure there is a sca- lar B such that ¥( Mg) = B fig: Proving the above-described preliminary result is as follows. Consider any set B of positive, finite measure. If ¥(P) is nota constant multiple of Ps it must be that either Re [y( 7, or Im xi 7g has the property that its infimum on one subset of posi- tive measure C (contained in B) is greater than its supremum on a- nother subset of positive measure D. This means, in view of the fact that some translate of C intersects Dina set of positive measure, _ that there exist in B subsets of positive measure, A and aA,for which T,(P,AY Je) #P,,¥ /,- But this yields a contradiction, since the following computation shows that Ti(PAY B) = Pia¥/ pi T(P,aY 3) = TYP a }s TA Y( bs he) TY 2a W ul KK rj i) ee > TT rq SR 9 > -80- ¥( Janke) - ¥(Pa, fp) Pat ile. This proves that for every set E of positive, finite measure there is a vfig- pA , which, as already shown, proves the present proposition, scalar f such that Definition 16: Given a collection thd of real (or complex) hilbert spaces we define the direct sum ( ® A, ) of LAS to be Kel the family of all functions f on the index set I such that 1) £(«) is a member of Rk for every «in I, 2) £(«) is the zero vector of K, for all but an at most ee) 00 countable collection of «'s fa} ja? and > lec? converges. 7 i=l wel h is a hilbert space if we define the inner product of any two mem- bers, f and g, by (Eg) = 2_(Eoyfela))) wel ; This provides a frequently useful method of expressing a given hilbert space in terms of a sufficiently exhaustive collection of closed, mutually orthogonal subspaces, if no vector of the space is orthogonal _to every member of the collection. For instance, the complex L*( [0, 1] ) can be mapped unitarily onto the direct sum of a countable collection of mutually orthogonal one-dimensional subspaces corres- ponding to the family of functions . n = +00 inx e n= -@ -81- Also, hilbert spaces with various unusual properties are constructible by forming direct sums of suitably chosen collections of familiar hilbert spaces, | ‘Lemma 12: L(G, S, ) is irreducible under the combined family of operators it nee iP eS EeS’ Proof: This result is known, but the proof is included for com- pleteness. It will suffice to prove the lemma for the case where L(G, s, LB) is a real hilbert space. For in the case where L(G, Ss, .) is complex we can verify the lemma from its validity for L the real hilbert 2 R? space made up of those elements of L(G, S, ») which correspond to . real-valued square-integrable functions. If we should have L(G, 5,u) =f, @A,, where each of the two spaces ky and A, is carried into itself by the family in question, L 2 would be contained in one of these two sub- R But, A, being a complex hilbert space, 2,., 2 . R + iL, showing the irreducibility of L“(G, Ss, }). spaces, say A. AOL = L(G, 5, u), So let us suppose that L(G, Ss, 2.) is real and that L“(G, s, B) = A @A ; where neither K nor kK is a trivial space andeach is carried i 2 1 2 P into itself by iv J acG Ute j Thus we may find f in A and g E’ EeS* in hs such that both éxle(x)> 03 (= E) and ixle(x)> 05 are sets of posi- tive measure. Further, by Corollary 2 to Proposition 2 of Section 1, there is an s in G for which p(sE NF) is positive. Thus (T Ppt, Pag) =/Pa(s tt(s de) Lp lelete) duce) = [tls berate) ante) sE (\F >0. ~82- ~\ But, since sisinh, and-g isin h,, we have by definition of A, and hk, that T Prfe h, and P pee hy, so that from the assumption that h, is orthogo- ‘nal to h, | (TP pf, Pps) = 0, a contradiction. This concludes the proof. . We are now in a position to prove our generalization of that theorem of Loomis which we stated at the beginning of this section. Theorem 1: Let (G, $:5,7;A) bea projection-measurable group. Then § supports a left-translation invariant measure p, unique up toa scalar factor, such that for all Ein§& y(E) = 0 if and only if T(E) = 0. Further, A is expressible as the direct sum of a family of mutually orthogonal, closed subspaces ths with the following property: for every there exists a unitary mapping V, (unique up to a uni- y ; modular constant factor) of L(G, s, .) onto A. for which we have Va dale = Th and -1 Vv, MEY = Pp for every ain Gand Ein S. Assuming all of the other defining proper- ties of (G,d;5,17;A), the following three properties form necessary and sufficient conditions for the truth of the present assertion: 1) 1(S) is full, 2) m(S) is countably decomposable, 3) ¢(x)€é is strongly measurable relative to 7 for every & ink. Proof: We give the sufficiency proof first. We shall devote our- -83- selves completely to the proof of the assertions about the structure of ¢, 7, and f, since our assertion about measure is just a restatement of part of Proposition l. | Using the terminology of Proposition 4 let us define the following linear mapping T on a linear subspace of L(G, S,») to a linear sub- n space of A. For every finite sequence of scalars ix J A and every = 1 n finite collection of mutually disjoint sets of positive measure iz.3 i=l belonging to R let n n T( Sw f,)= 22 «ME. i=l i i=l First, we shall verify that T is uniquely defined by showing that if E, F, and EUF are in Rand ENF=d¢ then MEUF) =ME) +X(F). Remembering, from Proposition 4 that for A ins and B in M A(Ang) = 7 (A) A(B), we have MEUF) = TIEUF)\MEUF) (T(E) + W(F))MEUF) T(E)MEUF) + 7(F)A(EU F) ME) +A\(F). Next T is unitary on its domain of definition. Note that any par- ticular pair of members of the domain of T may be regarded as two linear combinations of the same finite collection of mutually disjoint elements of R; hence, if -84- |esy|] ? = woe) for all E in R and if, whenever E\F = (ME), MF) = 0, it must be that 7 (Tt, Tg) = tg) for any f and g in the domain of definition of T. But we know by Pro- position 4 that . [Mceyl ? = woe) for every E in R. And if E/)F =¢ for E and F both in R the following computation shows that (XE), A(F)) ( vanishes: (Me), Me) = (rte) Me), ) Me) = (Fre) 7(F)) ME), A) (TE NF) ME), MF) = 0. We can see finally that for alla in Gand AinS To g(a)T = T, and -1 T WA)T = Pp. For in view of Proposition 4, we have T, bes Kan Tt!) (az) T7'4(a) \(E) tyr t, -85- and Py be hang Tt AANE) vo lr(apyA(E) t'mayty . for allainG, AinS, and Ein R, Finally, since T is a unitary mapping into A of its domain of defi- nition and the domain of definition of Tis, by Proposition 4, dense in L(G, S,), T can be uniquely extended to a unitary mapping (which we continue to call T) of all L(G, Ss, 4.) onto a closed subspace of k, And, in addition, the continuity of all operators involved being assured, the following identities verified in the last paragraph for a dense sub- set can be extended to the whole of L(G, S, bh): ) Tgla)T = T, 2) T'MA)T = Py, for every a in G and A in Ss. Now, it can be seen from these identities that T [u4(c, Ss, »)d is invariant and irreducible under the combined families 24(a)3 og and fr(ays AeS* The invariance property comes from rewriting the two identities just obtained as: 1') g(a) = TTT 2') a(A) = TP,T and remembering that T is a unitary mapping of L(G, S, ») onto T Rates Ss, wy] . The irreducibility property comes from combining Lemma 12 with identities 1) and 2) of the previous paragraph. -86- We can now show that f is the direct sum of a family of closed, mutually orthogonal subspaces, ik, 5 ep for each of which the results of the preceding paragraphs are true. For, consider a maximal collection of closed mutually orthogonal ) subspaces of h all of which have the same properties as T [L.7(G,8, wl (Such a collection exists by the arguments of the preceding paragraphs and by the Maximal Principle), and call the direct sum of its members ke. | If hok is not the zero vector, then, first of all, we know from the definition of a projection-measurable group that 7(S) is full and countably decomposable on k and so, a fortiori, is full and countably decomposable if contracted to Rk . In addition, because TEVA [oak] is a subset of R for every E inS la in el we have for every E in 5 (ain G), € in Aok, and nin A that (™(E)E, n) = (€, 7(E)n) = 0 [(dlayé, n) = (€, [d(a)]™ 9) = (6, d(a7')n) =o], | i.e., T(E)(A@R) [d(a)(AOR)] is a subset of hoRtor all-E in S (a in G). Hence, if we contract all members oflé(eh eg and 7(S) tohok, (G, 4;5,77; ROR) is a true projection-measurable group, the other pro- perties being clear from the definition of (G, ¢;5, 7h). The methods of preceding paragraphs of this proof are clearly ap- plicable to (G, é:5,7;A@QA), and they yield the result that there exists a closed subspace with the same properties relative to ¢ and 7 as those possessed by T [L2(G,5, wl - But this contradicts the fact that Ris the -87- direct sum of our maximal mutually orthogonal collection of closed sub- spaces of Kk each having the same properties as T [u2\c, Ss, wd . Thus it is false to assume that Rok is not the zero vector, which means that A possesses the desired decomposition. This completes the sufficiency proof for the present theorem. Now let us give the necessity proof. We are assuming the exis- tence of a 7 -finite ,left-invariant measure on § such that p(E) va- nishes if and only if T(E) vanishes. Also, we assume the decomposition of A into a family of mutually orthogonal, closed subspaces {hk J a with the property that | 1) k = v, [L71G, 5, 0] and | 2) Vx is a unitary mapping of L(G, S,}) into A such that on A | Va Taq = Bla) and | 1 Vv, P oa “" = W(E). The other conclusions of the present theorem will not be needed for our necessity arguments. | | The fact that p is a 7-finite measure on § causes ip_J Bes? COR- sidered as an Abelian algebra of projections on L(G, Ss, wu), to be full and countably decomposable. (These facts were pointed out in the pro- cess of defining fullness and countable decomposability.) Hence, in vir- tue of the relation between ip,§ EeS and 1 (S), it must be that 1(S) is full and countably decomposable. Next, we wish to show that ¢(x)€ is strongly measurable relative -88- to 7 for every € ink - Because of the relation between Ty. and ¢(x) and the assumption that 7(E) vanishes if and only if p(E) vanishes, it follows that this will be accomplished.if we show that on each mem- ber of S (minus a subset of zero measure) Tif is a uniform limit of countably-valued step-functions for every f in L(G, S, p)«- Observe that by Proposition lan arbitrary neighborhood of the identity in the topology T has the form bx [ futsce) + HE) - 2p(xE E) <e, O<e< 2uN(E)< wo and that M(xE) + w(E) ~ 2u(xE/E) = |oon - Kon? au. As a result tT (We) is continuous in the topology T. Then, linear combinations of such characteristic functions being dense in L(G, s, py.) by Lemma 11 and qT. being a unitary operator on this space, Tf is continuous in T for all f in L(G, s, wu). Then if we remember that Lemma 10 decomposes any given E in S into an at most countable col- lection of measurable subsets of left translates of any given neighbor- hood of the identity plus a set of measure zero, it follows that if we remove a subset of measure zero from E Tf is a uniform limit of countably-valued step-functions on the subset remaining, As re- marked, this shows that g(a)€ is strongly measurable relative to 7r for every € in h, This completes the necessity proof and, hence, completes the proof of Theorem l, Theorem 2: Let (G, é;5,7;h) be a projection-measurable group such that for any E in § there is an a in G such that T(EN\ (aE)‘) = 0. Then G possesses a topology T such that 1) there exists a family of neighborhoods of the identity -89- equivalent to the defining family and possessing the property that its members belong to S and have positive, finite y-measure; 2) Gis a topological group in the topology T such that each neighborhood of the identity is contained in the union of a finite number of left translates of any other neighborhood of the identity; 3) 4(x)& is continuous in T for every € in A; 4) Gis a dense subgroup of a locally compact group. Proof: Conclusions 1, 2, and 4 of this theorem are a rewording of the conclusions of Theorem 3 of Section 1. But, since by definition of a projection-measurable group S is an invariant 7-ring and by Pro- position | pis a left invariant, 0 -finite measure on S, (G, Ss, p)isa left-invariant GMG, the hypotheses of Theorem 3 of Section Lare satis- fied, with one exception. The hypothesis which is unverified is that if E is any member of S for which p(E) is positive then there is ana inG for which p(aE ME) <p(E). But the new hypothesis made on a projection- measurable group in the statement of the present theorem is just pre- cisely equivalent to the desired hypothesis, since w e. have that for every E inS y(E) vanishes if and only if 7(E) vanishes. So all but conclusion 3 of the theorem in hand may be considered proved. Now if we compare the statements of Theorem 4 of Section 1 and of Proposition 1 we see that under the present hypothesis they impose the same topology in G. This combined with the fact that Lemma 4 shows ¢(x)& to be continuous in the topology of Proposition l,proves the third conclusion of our theorem, Remark 1: The somewhat tortuous methods used here to obtain Theorems 1 and 2 might appear avoidable if the following approach is taken. First, restrict the discussion to projection-measurable groups -90- of the type considered by Theorem 2, which, after all, is not really much of a restriction. Then construct on § the left-invariant measure ». such that p(E) = 0 if and only if T(E) = 0. Next, apply Theorem 3 of Section I to the resulting left-invariant GMc, so that one can consider the locally compact group which is the completion of the GMG. (Theo- rem 2 of the present section also falls out at this point.) Finally, de- . fine 7 on the Baire sets of the locally compact group just constructed, extend ¢ to all of the locally compact group, and apply Loomis's theo- rem, which would seem to make Theorem 1 of this section true,a forti- } ori. But there is one difficulty: there is no way of knowing whether all of S or merely a proper %-subring of Ss corresponds to the Baire sets of our locally compact group, so that in applying Loomis's theo- rem as above we may end up with the representation of a proper subalge - bra of 1r(S) rather than of the whole of 1(S). The author could sée no | way around this, with the result that he took the present approach, name- ly, that of generalizing Loomis's arguments. Remark 2: Itis necessary and sufficient for the truth of Theorem 1 that 7(S) be full and countably decomposable and that ¢(x)€. be strongly measurable relative to 7 for every & ink, yet the mutual independence of these three conditions, in the context of the other properties which define a projection-measurable group, may seem dubious. This inde- pendence actually exists, for we shall now give three examples of structures each one of which has all properties of a projection-measura- . ble group excepting exactly one of our group of necessary and sufficient conditions for Theorem 1. A different condition will be violated in each example, of course, . -91- Example I (Violation of countable decomposability): Let our abstract group G be the direct product with itself of the group R of real members under addition, i.e., G=RxR Let S be the family of Baire sets in G relative to the Euclidean topo- logy on G. To define the hilbert space k , we observe that G underlies the topological group formed by giving one copy R, of R the discrete topology, giving a second copy R, of R the Euclidean topology, and constructing the product of the two topologies on the abstract product- group R, x R, (=G); then, letting T and V be, respectively, the family of Baire sets of the latter topological group and the Haar measure on T, we define A to be the real L7G, T,V). As might be expected now, we define the representation ¢ of G on a group of unitary opera- tors on A by the relation: g(x) = Ty for all x inG. (TY, is the unitary translation operator on L(G, T,v) defined in Definition 14.) But, we define a 7-homomorphism T of 5 on an Abelian ring of projections on A as follows: for every Ein s TE) = Py, where Py, is the projection operator on A (=L“(G, T,v)) corresponding to the multiplication of a function square-integrable on (G, T, V) by the -92- characteristic function of E. The structure just exhibited certainly has all the properties of a projection-measurable group other than those three with whose inde- pendence we are concerned. A discussion of the last three follows. We can show ¢(x)& has the right measurability property for every € in kh . Infact, let us consider Gas the abs™:;..ct group underlying the product topological group R, x R,, R, being the discrete reals and R,, the Euclidean reals. We see that since ¢(x) = TY gis representation of Gona group of unitary operators on the hilbert space L(G, Tv ). As shown in the example following the definition (Definition 8) of strong measurability relative to a7 -homomorphism, ¢(x)§ is strongly mea- surable relative to any @-homomorphism of T onto an Abelian ring of projections on L(G, Tv ). Since the domain of 7 contains T, it is now clear that ¢(x)€ is a strongly measurable function relative to 7. Next, 1 (S) is full since it contains all the projections on L(G, T,7) generated by characteristic functions of sets of positive, finite 7/-measure. * Since E is an arbitrary member of S, we must justify the last definition by showing that ENTCT for every E in S. We obtain our result by applying the following asser- tion first to T and then to §: the family of Baire sets of the product group G, x Gs generated by two locally compact topological groups, G, and G, if identical with the 7-ring generated by the collection of all cdrtesiat ‘products of the form By x B5, where B. is an arbitrary Baire set of G.. (See Theorem 51.C of (1).) Applying this assertion to T, we see that T consists of all sets of the form .W {x.} x E., where for alli x, is an element of and E, is a Baite subsét of R relative to the Euclidean topology. And, applying the quoted result to S, it follows that for every F in S every section of the form FP. is a Baire subset of R, relative to the Euclidean 1 topology. Thus for allEinS ENTCT. -93- But T(S) does not have the desired property of countable decom- posability. For any set in S having positive Haar measure, con- sidered as a Baire set in the Euclidean topology, contains an uncounta- ble collection of mutually disjoint sets having positive 7-measure and belonging to S, since positive V-measure is assigned to any set in G of the form fx} x E, x being any member of R and E being any Baire set in the Euclidean topology having positive Haar measure. 7 (E) f 0 for every E in S such that E belongs to T and WE) is positive, so it follows that there exists an F in S such that 7(F) contains uncountably many disjoint, non-zero projections of 7 (S). | Example II (Violation of strong measurability property): Let us take as G the additive group of the reals, R, and let us take as S the family of Baire sets relative to the Euclidean topology, or, what is the same thing, the family B of all Borel sets relative to the Euclidean topology. Defining m to be the Haar measure on B, i.,e., Borel mea- sure, we shall take AK to be the direct sum of a collection of continuum kR- Ox many copies of the real L7(R,B, m). If we write 2 «eR * and define for any € in A the projection of € on Lig’ to be ou? then for every € in K , Ein S(=B), and x in G{=R) (TIE)E), = Prey and (OIE, = TE a) (P,. and TS are the operators defined in Definition 14 above, E (G, 5, m)(=(R, B, m)) being after all a left-invariant GMG.) -94- Noting, among other things, that (R, B, m) is a left-invariant GMG, that feb os of projections on L?(R, B, m), and that a~pT, is a representation of is a full, countably decomposable Abelian algebra G on a group of unitary operators on L2(R, B, m),we see that the struc- ture just described has ail the properties of a projection-measurable group. Properties that may be lacking are the truth of the commu- tation relation: | d(a) T(E) = 7 (aE)¢d(a) for all a in G and all E in S and the strong measurability property of B(x) E. That the commutation relation holds is seen as follows: for every ainG, Ein Ss, é inh , and KinR (7(aE)é) P ii} aE ex T(LayPETa®((ata)-a) Ti_ayPp(F()8)( , 4a) Tay ENA) (ay) | (d(-a) T(E) d(a)é), But there exists an element é of A such that d(x) is not strongly — measurable relative toT. §€ is defined by the condition that for every in R _ 0, «#1 Ey = f, wx=1 where f is any element of L(G, s, m) such that lle ll =1. We observe from the definition of ¢ that -95- (dla)E, d(b)E) = 8, so that . listaye - spell = yz as long as a #b. Now let us look at the definition of strong measura- bility relative to 1 (Definition 8) and note that by our present definition of 7 if T(E) = 0 then E has zero Borel measure. If d(x)E is strongly measurable relative to jT,then on a set A having positive Borel mea- sure (hence an uncountable set) ¢(x)& is the uniform limit of a counta- ble sequence of countably-valued step-functions. Thus, there is a countable collection, F, of vectors of A such that {dta)e | aeal belongs to the closure of F. But this means that for every a in A there is an Ns in F such that Iloare - aS But the fact that Ilotayé - d(o)ell = 12 when a # b means that no two n,'§ can be the same vector; so in, 3 acA must be countable, A being an uncountable set. This conclusion con- tradicts the deduction that {n,§ a is a subset of the countable set F, eA which shows the falsity of the assumption that ¢(a)& is strongly mea- surable relative to T. Example III (Violation of fullness property): Let G be the real num- bers under addition, ‘S be the family of Baire sets in G relative to the Euclidean topology, and h be the hilbert space which is the direct sum of two copies of L(G, 5, m), where m is the Haar measure on s (Borel mea- sure). Further, letting (E,n) be a generic member of k. we can define a -96- go -homomorphism 7 on § to projections on A by the equation 7(E)(E, 0) = (Pp, 0) and a representation on G to unitary operators on k by the equation d(x)(E, n) = (T £0). (Pp and T, are the operators on L(G, S, m) defined in Definition 14.) Now to describe the structure that has been set up. To begin with, the fact that TP, = Plats makes it clear that ¢(x)7(E):=7(xE)¢(x). Since by reasoning used in Example 1 TLé is strongly measurable relative to T for every & in L(G, Ss, m) and ipsa is countably decomposable, then ¢(x)(€, n=(TL§, n)) must be strongly measurable relative to Tf for every (€, n) ink > and 1 (S) is a countably decomposable Abelian ring of projections. However, (S) is not full, for any element of h having the form (0,7), 4 } 0, is annihilated by every member of m8). Thus we have shown for a projection-measurable group the mutual independence of the hypotheses that a) 7 (S) be countably decomposable, b) 7 (S) be full, c) d(a)& be strongly measurable relative to 7 for every Having derived and discussed our generalization of Loomis' theorem, we propose now to give the actual proof that the latter is a consequence of our purported generalization. To accomplish this we need the following lemma and proposition, Lemma 13: Let G be a locally compact topological group and s be the family of all Baire sets in G. Suppose there exists a non-zero G-homomorphism 77 of S onto an Abelian ring of projections ona hilbert space hk, and suppose there is a representation ¢ of G on a group of uni- tary operators on hk such that -97- w(aE) = d(a”'r(E)d(a) for alla inGandall EinS. If ¢(x)€ is continuous on G for all € inh ; and if His an open Baire subgroup of G, then for every A in HONS T(A) = 0 if and only if A has Haar measure zero. _ Proof: For every € in A tet us define the measure. rhe on S by the equation | m,(E) = (W(E)E, €). We see that (G, Ss, m¢) is a @-finite measure space, so that (GxG, Sx s, x m™ ¢) is a measure space for which the Fubini theo- rem is valid. (y is the Haar measure on Ss.) Further, for every A in HAS, (s7'Rs)(a x H) is a member of S x § by Proposition 1 of Section I. (Let us remember from Section I that S(x, y) = (x, xy) and R(x, y) = (y, x).) As a result, we have by the Fubini theorem and Proposition 0 of Sec- tion J that A mp («1A )H™) due) = a ieee dyu(x) “L. be, y) dl # mete, eg (ST IRSMA x HOY) Uh x m_)Gs y) »([is"'Rsy(a x ]¥) am, ty) oN od a @) = fal BAY” *) ame (y) -98- [as dm, (y). H And, since the fact that A belongs to Hf/S means “La Nuh | _ -1 m,(x AiJH °) du(x)=/ m,(x A) dp(x), we have the following identity true for all A in H/M§ and all € ink . -1 / -l, 4. me (x “A) du(x)= / wlAy -) dmg(y).. H H Using this identity, we show first that if A belongs to HMS and (A) = 0 | then w(A) = 0. s being the 7-ring of Baire sets, it follows from the definition of S that each is contained in the union of an at most countable number of compact sets, so that every member of S is contained in the union of at most countably many left cosets of H. This means, in particular, that there must exist an E in § such that EN x H =E for some x and T(E) = 0.2. But since T(x, E) = 6x, r(E)b&,"') , x 'E is a member of H{5 on which 7 does not vanish. As a result, we know that for some EG in A me (M) = (7(M)&., E,)) does not vanish for every Min Hf\5S. Yet if 7{A) 29 for some A in H(MS, then me (TA) = (TETANE,, €) -99- (Sle) TIAN” )E,, 0) 0 J we (x7 ta) dy(x) H ° = 0. & for all x in H, so that (Ay!) dim, (y) H oO The fact that ™M¢ is not identically zero on H/)5 can now be brought re) to bear with the consequence that p(Ay') = 0 for at least one yin H. Thus, since right-translation maps sets of Haar measure zero onto sets of Haar measure zero, H(A) = | Now suppose that for some A in H/\5 B(A) = 0; then we can show that T{A) = 0. If p(A) = 0, then wlAy") = 0 for all y, so that for all € in m, (x7 'A) dulx) = / p(Ay” ‘) dm,(y) H m,(y) = 0. mt me Since we know that this holds, that every neighborhood of any point in G contains an open Baire set of positive measure, and that H is open, we can conclude that for every € in £_() = mp (x”'A) - 0 for x on a dense subset of H. Consequently, if we can show that fe (x) -100- is continuous at x =e, m,(A)= 0, But the identity 1 (xA) = d(x”) 1(A)¢(x) and the continuity of ¢(x)— prove the continuity of f at e: [igs - egCe)! = tmGmaye, 6) - (meade, él = [(r(A)d(x)é, d(x)€) - (T(ADE, EDL eg lrO ene - 6), 6608) + [(TAIE, dE - £)I 4 lhrrayideeré - dleréyll- | dxrell + fircarell- beare - o(epll é2ffell-lldooe - deréll . So m,(A) = 0 for all — ink, which in view of the identity m,(A) = (IMA) &, 6) = Irayell? means that ?(A) = 0. ‘Thus we have shown that for every A in HMS | w{A) = 0 if and only if T(A) = 0, which is the desired result. Proposition 6: Let G be a locally compact topological group and S be the family of all Baire sets in G. Suppose there exists a non-zero Tg -homomorphism 7 of S onto an Abelian ring of projections on a hilbert space h, and suppose there is a representation ¢ of G on a group of uni- tary operators on h such that -101L- T(aE) = g(a”') m(E)¢(a) for alla in Gand all Ein®. Then it d(a)&é is continuous on G to for every g in h 1(S) is countably decomposable. Proof: (Unless there is a statement to the contrary, we shall understand ''set'' to mean ''Baire set" wherever it occurs in the follow- ing.) The proof can be reduced to showing that, under the present hypo- theses,for every E in Ss T(E) = 0 if and only if the Haar measure on Eis zero. The reduction is possible s since the truth of the preceding assertion permits us to set up a bi- unique union- and intersection-preserving mapping ™(S) onto a countably decomposable Abelian ring of projections on L(G, Ss, »w), p being Haar measure. For, to begin with, mapping every element of Ss into the projection on L(G, Ss, wt) generated by its characteristic function is a 7g ~homomorphism of S onto an Abelian ring of projections A on L(G, S,»). Combining this with the hypothesis on 7, it follows that m4(S) and the last-mentioned Abelian ring of projections are maps of s under %-homomorphisms relative to each of which zero has the same inverse image. This fact, since the common inverse image of zero is the family of all Baire sets of Haar measure zero, makes the mapping of 1(S) onto A which sends 1(E) into the projection on L(G, S, w.) cor- responding to the characteristic function of E a biunique, as well as operation-preserving, mapping. And, as pointed out in the definition of countable decomposability, Abelian rings of the sort of A are countably decomposable. Thus, because of the relation it permits us to set up between T(S) and a countably decomposable Abelian ring, proof that -102- if and only if . y(E) = 0 yields the proof of the present lemma. To implement our argument, it will be useful to construct an open subgroup H of G which is a Baire set and, hence, a set of positive -measure. If O is an open Baire set, then the sequence of sets 00 fof n=l defined by the recursion relation o? - (0? jo is an increasing sequence of open sets of positive measure and H = U) o” n=] _ isa subgroup of G which is a set of positive measure. We prove that oO” (and so H) is a Baire set by induction. Since (G, 8, we) isa genera- lized measurable group, we know by Corollary 1 of Proposition 2 of Section I that lx 'o No is an S-measurable function on G, whence o* = fal u(x7to Nov} ? o3 is an S-measurable set, i.e., a Baire set. Further, wb to? on} is an S-measurable function and o8 th. (O0 = flee to™No-}> of if O” is S-measurable, i.e. a Baire set, and if we again use Corollary 1 of Proposition 2. Hence, ont t is a Baire set. Using the Principle of Finite Induction, every Oo”, and so H, is a Baire set. Now, since the family of all distinct left cosets of H constitutes a disjoint dissection of G into open sets and since every Baire set is contained in the union of at most countably many compact sets, every -103- member of S intersects. at most countably many left cosets of H. This means, in view of the invariance of » and the fact that ma) = d(a”')T(E)d(a), that proving . W{E) = 0 if and only if K{E) = 0 for all E in S is equivalent to proving the same assertion for all E in HS. But now the lemma just proved (Lemma 13) applies so that the present proposition can be considered proved. We can now derive a theorem which is essentially Loomis! theo- rem. The only differences are: 1) a correction in his statement about the uniqueness properties of the mapping described and 2) the addition of the necessary requirement that the Abelian ring of projections in- volved be full. Theorem 3: Let G be a lacally compact topological group, S be its family of Baire sets, and » be the left Haar measure on S. Suppose there exists a non-zero 7-homomorphism 7 on S to a full Abelian ring of projections on a hilbert space A , and suppose that there exists a strongly continuous representation ¢ of G on a group of unitary opera- tors on A naving the property that 7(aE) = ¢(a7')m(E)d(a). Then his expressible as the direct sum of a family of closed subspaces ik, fa such that: 1) For every « there exists a unitary mapping V, of L(G, S, ») onto kK {unique up to a unimodular constant multiple) with the property that for every a in G and every AinS -104- Vx 'dla)¥, = T, and V, MAI, = Py, 2) Every fis invariant and irreducible under the com- bined families f(a) $ acG and {mays Aes’ Proof: In the light of Theorem 1 of this section and the fact that the left-invariant measure on S$ is unique up to a constant multiple, we need only verify that (G, 6:5, T;h) is a projection-measurable group. But this quickly reduces to showing that for every € in kh A(x)E is strongly measurable on S relative to 7’ as we can see. For, to be- gin with, Ss, the family of Baire sets of G, is shown in the remarks following the definition of a generalized measurable group to be invari- ant, Also, the definition of a strongly continuous representation and Proposition 6 of this section yield the result that 7(S) is countably decomposable. And the truth of all other properties, with the excep- tion of the strong measurability property desired for ¢, are explicit in the hypotheses of the present theorem. To prove that ¢ has the desired strong measurability property, we shall remember that every set of positive measure is the union of an at most countable collection of mutually disjoint sets of positive, finite measure. It follows that we need only consider sets of the latter type in proving that ¢(x)€ is strongly measurable relative to 7 for all €. Now, since the Baire sets are generated by a family of compact sets, every Baire set is contained in the union of a countable collection of compact sets. Also, ¢(x)€ is continuous and so uniformly continuous -105- on each such compact set. But this means that ¢(x)€ is the uniform _ limit of a sequence of countably-valued step-functions on any given Baire set. Thus forall € 6(x)€ is by definition strongly measurable relative toT. This concludes the proof of the last result of this section. -106- ' Ill. CHARACTERIZATION OF SEPARABLE TOPOLOGICAL GROUPS POSSESSING A MEASURE In the present section we wish first to show that any separable topological group G possessing a certain kind of measure has as its completion a locally compact group G' such that G'-G contains no sets of positive Haar measure, and conversely. More specifically, we shall prove the following theorem: Theorem 1: Let G be a separable topological group. Let m be a o-finite measure on the o-ring generated by the family of all open sets in G such that: 1) m(aE) = 0 if and only if m(E) = 0, 2) { x{m(xE OE)> of contains a neighborhood of the iden- tity for every E such that m(E) > 0. Then the existence of m is necessary and sufficient for the existence of a locally compact topological group G' with the property that the group uniform. space of G!' is the completion of the group uniform space of G and G' - G contains no sets of positive Haar measure. The other purpose of this section is to determine necessary and sufficient conditions that a separable topological group shall be locally compact, Infact, the following will be proved: Theorem 2: Let G bea separable topological group. Then neces- sary and sufficient conditions that G be locally compact are that: 1) there be defined on the o-ring S generated by the open sets of G a o-finite measure m such that all left-trans- lates of sets of measure zero are sets of measure zero, 2) the group uniform space of G be complete. -107- We shall proceed to the proof of Theorem 1, beginning with several topological lemmas. Lemma 1: Given any family L005 vey of open subsets of a sepa- rable topological group G there is an at most countable subfamily . n <4 0, | “iJ Ged such that — nso Uo, = LJ 0x;. i=l xe I Proof: Since G satisfies the Second Axiom of Countability, there is a sequence of open sets such that for each i there is an & with the property that N, Co, and for eache It follows that if Oy is any 0. containing N,, then i 00 n fe) at, is the desired subfamily of the family fo, xel” Lemma 2: If Gx Gis the cartesian product of two copies ofa topological group G, each endowed with the same separable topology, then the resulting product topology To xG on Gx Gis separable. -108- Proof: By definition { M x NCG x GIM, Nareopen} is a basis for T6xG’ If n< 0 i=l {oi} is an at most countable basis for the topology on G, then, clearly, is an at most countable basis for T . ; GxG Lemma 3: If § is the o-ring generated by the family of open sets of ay separable topological group G, thenS x5 is an invariant @-ring (in the sense of Séction I).. Proof: From the definition we see that it is required to show that the transformation o (x, y)—(x, xy) simply permutes the members of § x § among themselves. This will be clear if we show that § x S is the o-ring generated by the family of all sets open in the product topology, for Gis a homeomorphism relative to the product topology. For, if O is the o-ring generated the family of all sets open GxG in the product topology on Gx G, then, first, CG xaP: where P is the o-ring generated by all cartesian products of pairs of open sets inG. Next, since every set open in the product topology is the union of a collection generators of P (an at most countable collection by Lemma l), -109- As a consequence, OG xG P. But P = §x 5, for, as shown by the lemma immediately succeeding Definition 1 of Section I, if Ris the o-ring generated by fo} gel ; then X x X is the o-ring generated by | . | 1S * Co ach Bel" Hence, wal wal th O GxG concluding the proof. Proposition 1: Let m be a o-finite measure on the o-ring 5 generated by the open sets of a separable topological group. Further, let m be such that for every a in G and E in § m(aE) vanishes if and only if m(E) vanishes. Then there exists a left-translation invariant o-finite measure ». on § which is equivalent tom, Further, G can be given a topology such that a base of neighborhoods at the point y is the family of all sets of the form: y 4x|p((xE - E)U(E - xE))<e, 0<e<2n(E) <eo} - In this topology, Gis a topological group, and the group uniform space of G has as its completion the group uniform space of a locally compact group G' such that G' - G contains no Baire sets of positive Haar mea- sure. Infact, if $ is the family of Baire sets of H and ty is the Haar measure on s, then Hf\SCs and w(HE) = QE) -110- for every E in s. Proof: We combine Lemma 3 of this section with Theorems 2 and 4 of Section I, assuming we know that for every a in G there is a set E of positive finite measure such that m(aE (\E)<m(E). But the latter is true because the topology on G is Hausdorff, sets open in the given topology have positive measure, and mis a o-finite measure. We can now show that if we strengthen the hypotheses on the measure m of Proposition 1 then the given topology on the group G of that proposition is equivalent to the constructed topology. However, we need the following preliminary result. | Proposition 2: Let m be a o-finite measure on the O-ring gene- rated by the open sets of a separable topological group. Further let mhave the following properties: 1) m(aE) = 0 if and only if m(E) = 0 for every a in G and every measurable set E, 2) { xlm(xE (\E)> o} contains a neighborhood of the iden- tity for all E of positive measure. Then the topology in which a generic neighborhood of an element y is given by y { xlu[ Gz - E)\U(E - xE)| <e, 0<e<2u(E)< eo} is equivalent to the given topology. (Here pis the left-translation in- variant o-finite measure asserted to exist and to be equivalent to m by Proposition 1.) Proof: As shown in the proof of Proposition 1, (G, $,m) isa GMG, if by S we mean the o-ring generated by all open sets of G. Then, in the terminology of Section I, (G,5,y) is a left-invariant GMG, ~lll- which is called a measurable group in (1). Now by Theorem 62.A of (1) for every set of the form M = { xin | (= -E)U(E - =) <e€, O<e <2u(E)<eo} there exists in every set of positive y-measure a subset F of positive -measure such that | FF7'cCM, so that {x nosr F<} CM, As a result, since we also know that Fala (Geo - E)U(E - xE)]<e, 0<e<2n(E)< co} = {acute QE)>,(E) - 5 >oh {xine ME)> 0} and thaty is O-+finite, we can prove the present proposition by showing that every neighborhood of the identity contains a set of the form {aclu NE)>0, 0<u(E)} and every set of the form {xx|u(xE NE)>o, 0<n(E)} contains a neighborhood of the identity. (In fact, m being equivalent . to yp, we can substitute m for p in the preceding assertion.) But ac- tually we need only demonstrate the former, since by hypothesis we have that for every E of positive measure {xtm(xz NE)> of ‘contains a neighborhood of the identity. | To show that every neighborhood of the identity contains a set of the form {xIm(xE(E)>0, 0<m(E) } we must first note that every neighborhood of the identity has positive m-measure. If Nis an arbitrary neighborhood of the identity and Eis any set of positive measure then EN contains E and, by Lemma 1,.is a countable union of left translates of N. Thus a countable union of left -l12- translates of N has positive measure, so that some particular left translate of N has positive measure. But, in view of the assumption that any left translate of a measurable set has zero m-measure only if the original set has zero m-measure, N has positive measure. - Now since for any neighborhood of the identity O there is a neighborhood N such that NN“'co, {ajm(cv (\N)>0} NNT Co. Inasmuch as we know that m(N) is positive, we have shown that an arbi- trary neighborhood of the identity contains a set of the form {x | m(xE()E)> 0, 0<m(E)} thereby completing the proof of this proposition. We can now see that Propositions l and 2, when taken together, yield the sufficiency proof for Theorem l. Next we obtain the results requisite to the necessity proof for Theorem l. Lemma 4: Let H be a dense sub-group of a locally compact topo- logical group G such that G - H contains no sets of positive Haar mea- sure. Let B be the family of Borel sets in G, and let p be the Haar measure on B. Then the function v on the o-ring -H(\B defined by the equation v(H{\E) = w(E) for all E in B is a left-translation invariant measure on H(\B. Proof; First of ali, ¥ is a uniquely defined function on H(\B. For if any pair of sets, E and F, in B are such that -113- HOE=H(F and H(E) # WF) then (EQ F°)U(FMES) is a measurable subset of G - H and has posi- tive measure, a contradiction of one of the assumptions on H. Second, y is completely additive. Suppose 1B: a}; =1 is a mutually disjoint collection sequence of sets in H(\B, and E, being members of B. Then every E. NE, has p-measure zero, each being a sub- set of G-H. As a consequence, C2) 00 > wE,)=H( U ED). n=l n= so that, by definition of By 00 00 » (HNE,) = U ,). n=l n=l | Hence, 20 co 2 wHME,)=v( U (Ne), n=l n=l proving the countable additivity of v. As to the left-invariance of ¥, observe that for any element a of H and any E in B, y fac =) = v (H{\aE) p{ak) p{E) vy (H[{)E). -1l4- Lemma 5: Given a locaily compact topological group G, let B be the family of all Borel sets of G, and let H be a subgroup of G separable in the relative topology. Then H{\B contains the o-ring generated by the collection of all subsets of H open in the relative | topology. . Proof: Observe that any x in H, considered as a point in the topological group G, possesses a family of neighborhoods which is made up of members of B and is equivalent to any other family of neighborhoods of that point. As a result, by the definition of the topo- logy on H, x, considered as a point in the topological group H, has the same property, except that H(|B plays the role played by B in the first case. Thus every open set in H is the union of a collection of open sets contained in H()8. Since by Lemma 1 and the separability of the topo- logy on H this collection can be taken to be countable, every open set in H must belong to H ‘a B. Thus the o-ring generated by all open sub- sets of H open in the relative topology on H is a subcollection of H(1B. Proposition 3: Let H be a dense subgroup of a locally compact topological group G such that H is separable in the relative topology and G - H contains no sets of positive Haar measure. Then there exists a measure Y on the O-ring generated by the family of all open sets in H which satisfies the hypotheses made on the measure m of Theorem l. Proof: We shall prove this proposition by showing that the restric - tion to S of the measure Y defined by Lemma 4 has the desired proper- ties. (This restriction is possible since, by Lemma 5, SCH(1B -115- and since H N B is the original domain of definition of v.) But we see from Lemma 4 that V is a left-translation invariant measure on S, from which it is obvious that for alla in Handall Ein5 waE) = 0 if and only if V(E) = 0. Further, every set of the form {xeHly (xE N=)>0} ; where V(E)>0, contains a set which is a neighborhood of the identity in the relative topology on H, for we know from the local compactness of G and from the continuity on G of p(xE OF) for every set F in B that if p(F)>0 {xeG] Her MF)>o} contains a neighborhood of the iden- tity. (The continuity of p(xF[]F) is proved on p. 50 of (7).) Since, finally, it is clear from the definition of V that itis a o-finite, we see that V’ is a measure on § having all the properties assumed for the measure m of Theorem 1. Having proved Proposition 3, we see that we have the necessity proof for Theorem 1, which means that we have disposed of the latter. completely since the sufficiency proof is contained in Proposition 2. Remark: One of the assumptions on the measure m of Theorem l was that for every E for which m(E)>0 the set {xim(xE (\E)> of must contain a neighborhood of the identity. Now, it might appear that some separable topological group satisfying the conclusion of Theorem 1 could be such as to permit a measure having all the pro- perties assumed with the exception of the one in question. But in actuality this seeming restrictiveness does not exist. Suppose that on the O-ring s generated by the family of open sets of a separable topological group H there exists a o-finite measure m and that, in addition, H is a dense subgroup of a locally compact topo - logical group G such that G - H contains no Borel sets of positive Haar ~ll6- measure, Then since Lemmas 4 and 5 apply to H we can use an argu- ment occurring in the proof of Proposition 3 to show there is a O-finite left-translation invariant measure Y on § such that {xeH| WxEf) E)> of contains a neighborhood of the identity whenever V(E)>0. But, since Sis an invariant o-ring ,(H, S,V) and (H,5,m) are both GMG's, so that by Proposition 7 of Section I m and Y are equivalent measures on S. It is now obvious that whenever m({E)>0 {xetilm(xE[\E)>q contains a neighborhood of the identity. Thus we see that the apparent restrictiveness of assuming this property for m in the statement of Theorem 1 is indeed nonexistent. Now, we proceed to the proof of Theorem 2, which will charac- terize locally compact separable groups. Definition |: A function g on a uniform space G to a uniform space G' is uniformly continuous if and only if the following condition is satisfied. If Vg'is any neighborhood of the diagonal in G'xG', then there exists a neighborhood Ve of the diagonal in G x G such that for every (x, y) in Ve (g(x), g(y)) is in Vg'. | As we see from the definition in Section I of a neighborhood of the diagonal (Definition B of Section J), gisa continuous function on G to G', Lemma 6: Let E be a uniform space, and let f be a uniformly continuous function from E onto a complete uniform space U. Then there exists one and only one function of f from the completion of E, E, onto U such that a) f is continuous on E to Uz b) f=fonE, Further, f is uniformly continuous on E. -117- Proof: This is a special case of Theorem III of (8). Lemma 7: Let the hypotheses of Proposition | hold. Every neighborhood of the identity in H,considered as a separable group,con- tains a neighborhood of the identity in the topology defined on H by Proposition 1. | Proof: For one thing, by Lemma 1 of this section, H is the union of a countable number of left-translates of any neighborhood of the iden- tity in the original topology on H; thus, p being left-invariant and defined on all open sets, if pis zero on any neighborhood of identity in the origi- nal topology, it is identically zero, contrary to the definition of a mea- sure. Further, given any neighborhood N in the original topology there is a neighborhood M such that MM"'cn, and, p being O-finite, there is a measurable subset F of M such that O<y(F)<o. As a result of all this, it follows that N contains a neighbor- hood of the identity in the topology defined on H by Proposition 1 fou [(xF - FUE - xF)]<e, 0<e<2utF <a} C fxtnxF FP o} C {alee (\M)> of = MM CN. \ Lemma 8; Let the hypotheses of Proposition 1 hold. Let the group uniform space of H be complete. Then there is a uniformly con- tinuous function f from the group uniform space of the locally compact group H of Proposition 1 onto the group uniform space of H. f has the further property that the restriction of it to His the identity mapping on H to itself. -118- Proof: In view of the preceding lemma, the identity mapping I on the abstract group H to itself is a uniformly continuous function on H, considered as the group uniform space of the topological group formed from it by Proposition 1, to the complete group uniform space of H in its original group structure, . Further, by Proposition | the group uniform space of the locally compact group H of that lemma is the completion of the former of these two uniform spaces. But now we see that the hypotheses of Lemma 6 are satisfied, so that the assertion of the present lemma follows, Lemma 9; Let H be the locally compact topological group con- structed in Proposition 1. Then there is a sequence n< 00 {«,} i=l of compact subsets of H such that H = i K. i 1s 1 Proof: Let K be a compact Baire subset of Hf such that K = K7! and K has positive Haar measure. Then from the continuity of the mapping -1 f: (x, y) +xy on H x H to H we see that {«"} =1 is a sequence of compact Baire sets of positive measure, so that is a subgroup of H which is a union of compact Baire sets. Now let us use the terminology of Proposition and let us observe that -119- H = ad agl, {a4} ger is a family of mutually disjoint left cosets of the Baire subgroup L, where It. is apparent that H=H()\H ael and that MH (a gh) = Bagh) = E(L) 70. But since »(H) is positive and wis a O-finite measure, we know that H can contain at most countably many measurable subsets of positive H-measure, which caused I to be an at most countable set. Thus, _ 00 H= U aL i=t 00 00 =U at U ®) i=1 j=l 00 00 = U ¢( U a,x’) i=l j=l 00 = U C. n=l where all c.'s are compact, Lemma 10: Let H be a separable topological group such that its group uniform space is complete. Then H is not expressible as a count- able union of nowhere dense,closed sets, i.e., H is not of the second category. -120- Proof: Since H is separable, there is by definition a countable collection of open sets i+ ¢] fo}; . 1. such that every open set in H is the union of a collection of O,'s. As a result, any family of neighborhoods of the identity is equivalent to | the collection of all O;'s containing the identity. Now if where the C's are all closed and nowhere dense, then,because of these properties of the C's and the remarks of the preceding paragraph, the fol- lowing is true. There is a sequence of elements oO {x,} i=l and a sequence of open sets re) {op a1 such that every O; contains the identity, every neighborhood of the iden- tity contains some O;, for every i 97% 4 1)OG4 + 1p and i *%0:C( U cS, je? Then the family of all sets each of which contains some x,;OQ; as a subset is a Cauchy filter and, H being assumed a complete uniform space, must converge; so, there is an Xo in H such that Xo belongs to every x;0.. ~121- Hence, 4 -) tas io) n ms py pete i -~ a 18 18 rare a pond — Xam QO anv Lh Gn fo) rc rary u Cig «Cs for a contradiction. It can now be seen that it is false to assume H to be a countable union of closed, nowhere dense sets. Proposition 4: Let H be a separable topological group such that its group uniform space is complete and such that there exists a left- translation invariant measure of the o-ring generated by the family of all open sets. Then His locally compact. Proof: Let us apply Proposition 1 to H and consider the locally compact group H whose group uniform space is the completion of the group uniform space of H in the topology imposed upon it by Proposition 1. Let us obtain by Lemma 6 a sequence of compact sets in H, 00 {x,} i=l?’ such that Finally, let f be the function of Lemma 8 which maps fi continuously onto H in its original topology. -122- Now, since But each K, is compact, and f is continuous, so that £(K,) is compact for every i and, a fortiori, closed. H being the union of the £(K,)'s, it now is clear by Lemma 9 that at least one has interior. Thus for some i; there is a point y and a neighborhood M of the identity such that yMC4(K, ); ° whence ! MCy"'£(K, ). oO Finally, inasmuch as there is a neighborhood N of the identity such that NN cy, we have © NCNn7! CM ch kr, ). [e) Thus the closure of N is compact, being a closed subset of the compact set (y')E(K, ). Theorem 2: Let H be a separable topological group. Then neces- sary and sufficient conditions that H be locally compact are that 1) the group uniform space of H be complete, 2) there be defined a O-finite measure m on the o- ring S generated by the open sets of H such that m(aE) = 0 if and only if m(E) = 0 for every ain Hand Ein Ss. -123- Further, mis equivalent to the Haar measure on S, which is the family of Baire sets if H is locally compact. Proof: I. Sufficiency: As shown in Proposition | of this section there exists on Sa left-translation invariant o-finite measure which is equivalent tom.» Combining the existence of » with the completeness and separability of H we see that the hypotheses of Proposition 4 are satisfied, making H locally compact. Because § is clearly identical with the family of Baire sets if H is locally compact, p, a left-translation invariant Baire measure, must be a constant multiple of the Haar measure on s. Il. Necessity: By the second form of Theorem 7 of (8) every compact uniform space is complete. Since the group uniform space of a locally compact group is uniformly locally compact, it must then be complete. Hence, combining this fact with the fact that for. every locally compact group G there exists a left-translation invariant, o-finite measure on the family of Baire sets of G (identical with the o-ring generated by its open sets), we have the necessity proof for this theorem. The preceding theorem possesses a corollary which may be of some interest, namely: | Corollary: A (real or complex) separable Banach space B is such that there exists a left-translation non-singular measure on the o-ring generated by its open sets, if and only if it is of finite dimension, or, if and only if there exists a linear homeomorphism of B onto a (real or complex) Euclidean space, Proof: We observe first that if B is finite -dimensional itis linear- -124- ly homeomorphic to a Euclidean space, a well-known result. Next, if B is linearly homeomorphic to a Euclidean space, then it supports a left-translation invariant O-finite measure on the O-ring generated by its open sets. For if Eis real (complex), the n-dimen- sion ({(2n) - dimensional) Borel measure is well-known to be just such a measure on the o-ring generated by the open sets of E. Finally, suppose that the o-ring generated by the open sets of B supports a o-finite measure such that sets of measure zero are always translated onto sets of measure zero; then B must be finite dimensional. We shall prove this in two steps: first, by showing that B is locally compact, and, second, by showing that a locally compact Banach space must be of finite dimension, To show that B is locally compact we show that its group uniform space is a complete uniform structure and then invoke Theorem 2 to conclude that it is locally compact. But we can see that the group uniform space of B is complete, since in any metric space any Cauchy filter is the same as the filter of all sets each one of which contains ioe) > re) some member of the family of sequences {{a, ke ate =1 {a}, - being a fixed Cauchy sequence, Now, in order to prove that a locally compact Banach space has finite dimension and thereby finish the proof of our corollary, we shall show that Banach space which is not finite-dimensional is locally com- pact. To do this, we construct by induction an infinite sequence {x,} _. l of elements of B such that Hx = 1 for all n and Ix, - x1 >5 whenever m is unequal to n., k Suppose that we have constructed a finite sequence {x,} n=l with these properties, and the further properties that it is linearly in- -125- dependent. Then the set of all linear combinations of the x,,'S is closed. Next, ‘denoting the set of all linear combinations of the x's by M,, we observe that in virtue of M,. being closed and the fact that B is not finite-dimensional there exists a non-zero element X of B- such that IIx - ™, is not zero. So, if m belongs to M and is such k that then | %-m _ 1 =i “ecg Feo Ml >(—L—) Ix - m, | az - M l > . But now, combining the definition of M,. with the above inequality, we see that if we define x _ xX-m K+! Iz - mil thenll | =1 I. -x l, l fork +1. Thus the existence Met? ke LT MBP . of the sequence {x =] implies the existence of an x such that k+1 nen. k k+1 ix. $ n=1 has the same properties as Ix, | nel’ Applying the Principle of Finite Induction, we see that there does indeed exist a se- re) quence of (x, | n=l of elements of B such that | x, ll = 1for all n and | Xn 7 x, [>> whenever m is unequal to np. -126- As remarked above, this shows that if B is locally compact it must have finite dimension, which completes the proof of this corol- lary. Remark: It might be of some interest to point out the logical in- dependence of the two provisions of the criterion for local compactness given by Theorem 2. For, to begin with, it is clear from the above corollary to Theo- rem 2 that there exists a complete separable topological group which does not permit the existence of the sort of measure required by Theo- rem 2, On the other hand, there exists the desired kind of measure on the O-ring generated by the open sets of a separable topological group which is not complete. In this case, our example is found in Exercise 6 at the end of Section 62 of (1); here there is constructed a proper subgroup K of the real Euclidean plane E, such that E, - K contains no sets of positive Borel measure. Since all open sets have positive Borel measure, K is thus a dense subgroup of E,. Consequently, being a proper. subgroup of E,, K cannot be complete relative to the Euclidean metric and so cannot be complete considered as a uniform space; this is a consequence of the fact that in a uniform space determined by a metric every Cauchy filter is the family of all sets each of which contains a member of the 00 00 og family of sequences {{a,, } k= at n=P {ad k=l being some suitably chosen Cauchy sequence. But K is a separable topological group supporting a left-translation invariant measure on the o-ring generated by its open sets. For Kis clearly a separable topological group in the relative Euclidean topology. -127- Further, since the left-translation invariant Borel measure is defined on the 7 -ring s generated by the open sets of E,, Lemma 4 of this section shows that there is a left-translation invariant measure de- fined on the ¢-ring KNM5. However, K!S5 contains the 7 -ring generated by the subsets of K which are open in the relative topology. This verifies all the properties asserted for K and shows the in- dependence of the completeness condition in Theorem 2. -128- REFERENCES Paul R. Halmos, Measure Theory, (1950), New York. Einar Hille, Functional Analysis and Semi-Groups, (1948), American Mathematical Society Colloquium Publications, New York. Lynn H. Loomis, Note on a Theorem of Mackey, (1952), Duke Mathematical Journai, vol. 19, pp. 641-645, Béla von S. Nagy, Spektraldarstellung Linearer Transformationen des Hilbertschen Raumes, (1942), Ergebnisse der Mathematik und Ihrer Grenzgebiete, Berlin. Stanislaw Saks, Theory of the Integral, (1937), Monografie Matematyczne, second revised edition, Warsaw. Irving E. Segal, Decompositions of Operator Algebras, I and II, (1951), Memoirs of the American Mathematical Society, Number 9, New York. André Weil, L'Intégration dans les Groupes Topologiques et Ses Applications, (1953), Actualités Scientifiques et Industrielles, second edition, Paris. 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