ON LEFT-INVARIANT BOREL MEASURES ON THE

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1 Georgian International Journal of Science... Volume 3, Number 3, pp. 1?? ISSN c 2010 Nova Science Publishers, Inc. ON LEFT-INVARIANT BOREL MEASURES ON THE PRODUCT OF LOCALLY COMPACT HAUSDORFF GROUPS THAT ARE NOT COMPACT. Gogi Pantsulaia Department of Mathematics, Georgian Technical University, Kostava Street-77, Box 0175, Tbilisi, Georgia I.Vekua Institute of Applied Mathematics, Tbilisi State University, University Street - 2, Box 0143, Tbilisi, Georgia address: g.pantsulaia@gtu.ge Abstract We prove an existence of a left-invariant quasi-finite Borel measure on the product of an arbitrary family of locally compact Hausdorff groups that are not compact Mathematics Subject Classification: Primary 28xx; Secondary 28Bxx, 28Cxx Key words and phrases: Locally compact Group, Haar measure, left-invariant measure Let (X i,b i,µ i ) i N be a sequence of regular Borel measure spaces, where X i is a Hausdorff topological space. In [1] has been proved an existence of a Borel measure µ on i N X i (with respect to the product topology) such that if K i X i is compact for all i N and i N µ i (K i ) converges, then µ( i N K i ) = i N µ i (K i ). The goal of the present paper is to obtain a certain extension of this result for an arbitrary family (X i,b i,µ i ) of quasi-finite regular Borel measure spaces where X i is a locally compact Hausdorff group for i I. As a consequence, we are going to construct a left-invariant quasi-finite Borel measure on the product of locally compact Hausdorff topological groups that are not compact. If card(i) ℵ 0, then under such a measure can be considered a standard product (S) λ i or an ordinary product (O) λ i (cf. [2]) of the family (λ i ), where λ i is a Haar measure on X i for i I. Note that every locally compact group which is second-countable is metrizable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete. Hence, if card(i) > ℵ 0 and H i is second-countable for i I which is not compact, then H i stands a Polish group and following [3], there is a left-invariant quasifinite 1 Borel measure (S) λ i on the product of locally compact Hausdorff topological The designated project has been fulfilled by financial support of Grant: # Sh. Rustaveli GNSF / ST ) ). 1 A measure µ is called quasi-finite if there is X dom(µ) such that 0 < µ(x) < +.

2 groups (H i ) that are not compact such that if K i is compact in H i for i I such that then ((S) 0 (S) λ i (K i ) < +, λ i )( K i ) = (S) λ i (K i ). Here arises a question asking whether there exists a left-invariant quasi-finite Borel measure µ on the product of locally compact non-metrizable Hausdorff locally compact groups (H i ) that are not compact such that if K i is compact in H i for i I such that 0 (S) λ i (K i ) < +, then µ( K i ) = (S) λ i (K i ). Remark 1 Recall that a space is said to be second-countable if its topology has a countable base. More explicitly, this means that a topological space T is second countable if there exists some countable collection of open subsets (U k ) k N of T such that any open subset of T can be written as a union of elements of some subfamily of (U k ) k N. X is a completely regular space if given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F. In fancier terms, this condition says that x and F can be separated by a continuous function. X is a Tychonoff space, or T 3 1 space, or T π space, or completely T 3 space if it is both 2 completely regular and Hausdorff. Following Tychonoff s well known theorem, every product of arbitrarily many compact topological spaces is again compact. Since every locally compact regular space is completely regular, and therefore every locally compact( in particular, compact) Hausdorff space is Tychonoff, we claim that every product of arbitrarily many compact Hausdorff topological spaces is again completely regular. Definition 1 A Borel measure µ, defined on a Hausdorff topological space (X,τ), is called Radon if ( Y)(Y B(X) & 0 µ(y) < + µ(y) = and called dense, if the condition holds for Y = X. if sup µ(k)) K Y K is compact in X Definition 2 A family (U i ) of open subsets in (X,τ) is called a generalized sequence ( i 1 )( i 2 )(i 1 I & i 2 I ( i 3 )(i 3 I (U i1 U i3 & U i2 U i3 ))). 2

3 Definition 3 A Borel probability measure µ defined on X is called τ-smooth if, for an arbitrary generalized sequence (U i ), the condition µ( U i ) = sup µ(u i ) is valid. Definition 4 A Baire probability measure µ on X is called τ 0 -smooth if, for an arbitrary generalized sequence (U i ) of open Baire subsets in X, for which U i is also a Baire subset, the condition µ( i ) = supµ(u i ) U is valid. The following lemma plays a key role in our future investigations. Lemma 1 ( [4], Theorem 3.3, p. 42) Let X be completely regular topological space, µ be a Baire probability measure defined on the σ-algebra B 0 (X). Then (a) if µ is τ 0 -smooth, there exists a unique τ-smooth Borel extension on X. (b) if the space X is Hausdorff and µ is dense on B 0 (X), then µ admits a unique Radon extension on B(X). Let (E j,τ j ) be a family of Hausdorff topological spaces. We denote by ( E j,τ) the Tychonoff product of the family of topological spaces (E j,τ j ). Lemma 2 Let (E j,ρ j ) be a family of non-empty compact Hausdorff spaces and let µ j be a regular Borel probability measure on E j for j α. Then the product measure µ j is τ 0 -smooth and dense on E j. Proof. It is obvious that dom( µ j ) = B 0 ( E j ). Let (U i ) be an arbitrary generalized sequence of open Baire subsets in E j, for which U i is also a Baire subset. The latter relation follows that there exists a countable subset α 0 α and U α0 B( 0 E j ) such that U i = U α0 ( E j ) \α 0 and ( j)( j α \ α 0 (E j,ρ j ) is compact). By inner regularity of the Borel probability measure 0 µ j there exists an increasing family of compact sets (F k ) k N such that F k U α0 and lim n µ j (Fn ) = µ j (Uα0 ). 0 0 We set D n = F n j (α\α0 ) E j for n N. It is obvious that (D n ) n N is an increasing family of compact subsets in E j such that lim n µ j (Dn ) = µ j ( U i ). 3

4 It is obvious that (U i ) is covering of D n for every n N. Hence, using the definition of the generalized sequence of open sets in topological space, we can construct such a sequence (i n ) n N of indices of I that the sequence (U in ) n N will be increasing and D n U in for n N. We have µ j (Dn ) µ j (Uin ) for every n N. Hence µ j ( U i ) = lim n µ j (Dn ) lim n µ j (Uin ) µ j ( U i ). The latter relation means that the condition j ( U i ) = sup µ µ j (Ui ) holds. Thus, the measure µ j is τ 0 -smooth on E j. Let us show that the measure µ j is dense. Assume that j 0 α. It is clear that for the Borel measure µ j0 there exists an increasing sequence of compact subsets (F k ) k N in E j that lim µ j(f k ) = 1. k + Now it is easy that (F k \{ j0 }) k N is an increasing sequence of compact subsets in E j such that lim ( µ j )(F k E j ) = 1. k + \α 1 Lemma 3 Let (E j,ρ j ) be a family of non-empty compact Hausdorff spaces and let µ j be a regular Borel probability measure on E j for j α. Then there exists a unique τ- smooth and Radon extension of the Baire probability measure µ j from the σ-algebra B 0 ( E j ) to the σ-algebra B( E j ). Proof. By Tychonoff theorem E j is compact. Since E j is Hausdorff, following Remark 1 we claim that E j is completely regular. By Lemma 2 the Baire measure µ j is τ 0 -smooth and dense on E j. By Lemma 1 we deduce that there exists a unique τ-smooth and Radon extension of the Baire measure µ j from the σ-algebra B 0 ( E j ) to the σ-algebra B( E j ). We have the following lemma. Lemma 4 ( [5], Lemma 4.4, p. 67 ) Let (E 1,τ 1 ) and (E 2,τ 2 ) be two topological spaces. Denote by B(E 1 ) and B(E 2 ) (correspondingly, B(E 1 E 2 )) the class of all Borel 4

5 subsets generated by the topologies τ 1 and τ 2 (correspondingly, τ 1 τ 2 ). If at least one of these topological spaces has a countable base, then the equality B(E 1 ) B(E 2 ) = B(E 1 E 2 ) holds. Let us recall a definition of the standard product of non-negative real numbers (β j ) [0,+] α. Definition 5 A standard product of the family of numbers (β j ) is denoted by (S) β j and defined as follows: (S) β j = 0 if i α ln(β j ) =, where α = { j : ln(β j ) < 0} 2, and (S) β j = e ln(β j ) if ln(β j ). Let (E,S) be a measurable space and let R be any subclass of the σ-algebra S. Let (µ B ) B R be such a family of σ-finite measures that for B R we have dom(µ B ) = S P(B), where P(B) denotes the power set of the set B. Definition 6 A family (µ B ) B R is called to be consistent if ( X)( B 1,B 2 )(X S & B 1,B 2 R µ B1 (X B 1 B 2 ) = µ B2 (X B 1 B 2 )). The following assertion plays a key role in our future investigation. Lemma 5 ( [2], Lemma 1) Let (µ B ) B R be a consistent family of σ-finite measures. Then there exists a measure µ R on (E,S) such that (i) µ R (B) = µ B (B) for every B R ; (ii) if there exists a non-countable family of pairwise disjoint sets {B i : i I} R such that 0 < µ Bi (B i ) <, then the measure µ R is non-σ-finite; (iii) if G is a group of measurable transformations of E such that G(R ) = R and ( B)( X)( g)( ( B R &X S P(B) & g G ) µ g(b) (g(x)) = µ B (X)), then the measure µ R is G-invariant. Remark 2 Let (E j,ρ j ) be again a sequence of compact Hausdorff spaces and let (µ j ) be a sequence of non-zero regular Borel diffused finite measures with dom(µ j ) = B(E j ) for j α and 0 < (S) µ j (E j ) < +. By Lemmas 3, we claim that there exists a unique τ-smooth and Radon Borel extension λ of the Baire probability measure. A Borel measure µ j µ j (E j ) (S) µ j (E j ) λ is called a standard product of the family of finite Borel measures (µ j ) and is denoted by (S) µ j. 2 We set ln(0) = 5

6 Lemma 6 Let α be again an arbitrary infinite parameter set and let (α i ) be its any partition such that α i is a non-empty finite subset of the α for every i I. Let µ j be a quasifinite inner regular diffused Borel measure defined on a locally compact Hausdorff space (E j,ρ j ) for j α. We put τ i = i µ j. We denote by R (αi ) the family of all measurable rectangles R E j of the form R i with the property 0 (S) τ i (R i ) < such that R i is compact for i I. We suppose that there exists R 0 = R (0) i R (αi ) such that and For X B(R), we set µ R (X) = 0 if 0 < (S) τ i (R (0) i ) <. µ R (X) = (S) (S) τ i (R i ) = 0, τ(r i ) ( τ iri ) (X) τ i (R i ) otherwise, where τ ir i τ i (R i ) is a Borel probability measure defined on R i as follows τ ir i ( X)(X B(R i ) τ i (R i ) (X) = τ i(y R i ) ). τ i (R i ) Then the family of measures (µ R ) R R is consistent. Proof. Let R 1 = R (1) i and R 2 = R (2) i be two elements of the class R = R (αi ). Without loss of generality it can be assumed that 0 < (S) τ i (R (1) i ) < and 0 < (S) τ i (R (2) i ) <. We will show that µ R1 (X) = µ R2 (X) for every X B(R 1 R 2 ). In this case it is sufficient to show that µ R1 (Y) = µ R2 (Y) for every elementary measurable rectangle Y = Y i in R 1 R 2. Note here that under an elementary measurable rectangle Y = Y i in R 1 R 2 we assume a subset of R 1 R 2 such that Y i B(R (1) i R (2) i ) for every i N and, in addition, there exists a finite subset I 0 of I such that Y i = R (1) i R (2) i for i I \ I 0. For every i I and for every Y i B(R (1) i R (2) i ) we have The latter relation implies that τ i (Y i R (1) i R (2) i ) = τ i (Y i R (1) i ) = τ i (Y i R (2) i ). (S) τ i (Y i R (1) i R (1) i Hence we get µ R1 ( Y i ) = (S) ) = (S) τ i (Y i R (1) i τ i (Y i R (1) i ) = (S) 6 ) = (S) τ i (Y i R (1) i ). τ i (Y i R (1) i R (2) i )) =

7 (S) τ i (Y i R (2) i )) = µ R2 ( Since a class A(R 1 R 2 ) of all finite disjoint unions of elementary measurable rectangles in R 1 R 2 is a ring, and since, by definition, the class B 0 (R 1 R 2 ) of Baire subsets of R 1 R 2 is a minimal σ-ring generated by the ring A(R 1 R 2 ), we claim (cf. [?], Theorem B, p. 27) that the class of all those sets of R 1 R 2 for which this equality holds coincides with the class B 0 (R 1 R 2 ). Since restrictions of µ R1 and µ R2 to the class B 0 (R 1 R 2 ) coincide, and R 1 R 2 is a product of non-empty compact Hausdorff spaces, by Lemma 3 we claim that their Borel extensions coincide such that an extended Borel measure is unique, τ-smooth and Radon. The latter relation means that the family of measures (µ R ) R R is consistent and Lemma 6 is proved. Y i ). Let α be again an arbitrary infinite parameter set and let (α i ) be its any partition such that α i is a non-empty finite subset of the α for every i I. Let µ j be a quasi-finite inner regular continuous Borel measure defined on a locally compact Hausdorff space (E j,ρ j ) for j α. We denote by R (αi ) the family of all measurable rectangles R E j of the form R i with the property 0 (S) τ i (R i ) < such that R i is compact for i I. We suppose that there exists R 0 = R (0) i R (αi ) such that 0 < (S) τ i (R (0) i ) <. Definition 7 A Borel measure ν (αi ) defined on B( E j ) is called a standard (α i ) - product of the family of quasi-finite continuous inner regular Borel measures (µ j ) defined by the class R (αi ) if for every we have R = R i R (αi ) where τ i = i µ j for i I. ν (αi ) (R) = (S) τ i (R i ), Theorem 1. Let µ j be a quasi-finite continuous inner regular Borel measure defined on a non-empty locally compact Hausdorff space (E j,ρ j ) for j α. Let α be again an arbitrary infinite parameter set and let (α i ) be its any partition such that α i is a non-empty finite subset of the α for every i I, and let us suppose that there exists R 0 = R (0) R (αi ) such that 0 < (S) τ i (R (0) i ) <. Then there exists a standard (α i ) -product of the family (µ j ) defined by the class R (αi ). 7 i

8 Proof. For R R (αi ) and for X B(R), we set µ R (X) = 0 if and µ R (X) = (S) (S) τ i (R i ) = 0, τ(r i ) ( τ iri ) (X) τ i (R i ) otherwise, where τ ir i τ i (R i ) is a Borel probability measure defined on R i as follows τ ir i ( X)(X B(R i ) τ i (R i ) (X) = τ i(y R i ) ). τ i (R i ) By Lemma 6, the family of measures (µ R ) R R is consistent. We set ν (αi ) = µ R(αi ), where the measure µ R(αi ) is defined by Lemma 5. This ends the proof of Theorem 1. In the sequel we denote a standard (α i ) -product of the family (µ j ) defined by the class R (αi ) by (SC,(α i ) ) µ j If α i is a singleton of α for i I, then we use a notation (SC) µ j. Theorem 2. Under assumptions of Theorem 1, if each measure µ j is G j -left-and-rightinvariant, where G j denotes a group of Borel transformations of the E j for j α, then the measure (SC,(α i ) ) µ j is G j -left-and-right-invariant. Proof. We set G = G j. Let us show that the measure (SC,(α i ) ) µ j is G-leftand-right-invariant. Indeed, let g, f G and X B( E j ). If X is not covered by countable family of elements of R (αi ) then such will be gx f because the class R (αi ) is left-and-right-invariant, i.e., gr (αi ) f = R (αi ) for every g, f G. Hence, by the definition of the measure ((α i ) ) µ j we have (SC,(αi ) ) µ j (gx f) = +. Now let X be covered by the family (A k ) k N of elements of R (αi ). Then gx f will be covered by the family (ga k f) k N of elements of R (αi ). Hence, we get (SC,(αi ) ) j (gx f) = λ gan f((ga n f \ µ n 1 k=1 ga k f) gx f) = λ gan f(g((a n f \ n 1 k=1 A k f) X f)) = 8

9 λ An f((a n f \ n 1 k=1 A k f) X f) = λ An f((a n \ n 1 k=1 A k) X) f) = k=1 A k) X) = ( (SC,(α i ) ) µ j ) (X). By the scheme used in the proof of Theorem 2, one can prove the following assertion. Theorem 3 Under assumption of Theorem 1, if each measure µ j is G j -left-invariant, where G j denotes a group of Borel transformations of the E j for j α, then the measure (SC,(α i ) ) µ j is a G j -left-invariant. Observation 1. Under conditions of Theorem 1, the measure (SC,(α i ) ) µ j is Radon. Proof. Let 0 < ( (S,(α i ) ) µ j ) (X) <. It means that X B( E j ) is covered by any countable family (A n ) n N of elements of R (αi ) such that (SC,(αi ) ) j (X) = λ An ((A n \ µ n 1 k=1 A k) X). Since the measure λ An is Radon, we can choose a compact set F n (A n \ n 1 k=1 A k) X such that ( λ An ((An \ n 1 k=1 A ) ε k) X) \ F n < 2 n+1 for n N. Also, we can choose a natural number n ε such that k=1 A k) X) < ε n=n ε Finally, we get (SC,(αi ) ) µ j (X \ n ε s=1 F s) = k=1 A k) (X \ n ε s=1 F s)) = n ε k=1 A k) (X \ n ε s=1 F s))+ k=1 A k) (X \ n ε s=1 F s)) n=n ε +1 9

10 n ε λ An (((A n \ n 1 k=1 A k) X) \ F n )+ k=1 A k) X) ε n=n ε ε 2 = ε. Corollary 1 Let (H j,τ i ) be a non-empty locally compact Hausdorff group for j α which is not compact. Without loss of generality, we can assume that λ i is a left(right or left-right)-invariant Haar measure on (E j,τ i ) for which K j is compact in H j for j α and 0 < (S) j J λ j (K j ) < +. Then (SC) j J λ j is a quasi-finite left(right or left-right)- invariant Radon Borel measure on H j. References [1] LOEB, PETER A.; ROSS, DAVID A. Infinite products of infinite measures. Illinois J. Math. 49 (2005), no. 1, (electronic). [2] PANTSULAIA G.R., On ordinary and Standard Lebesgue Measures on R, Bull. Polish Acad.Sci., 73(3) (2009), [3] PANTSULAIA G.R., On a standard product of an arbitrary family of σ-finite Borel measures with domain in Polish spaces, Theory Stoch. Process, vol. 16(32), 2010, no 1, p [4] VAKHANYA N.N., TARIELADZE V.I., CHOBANYAN S.A., Probability distributions in Banach spaces, Nauka, Moscow (1985) (in Russian). [5] PANTSULAIA G.R., Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Science Publishers, Inc (2007), xiv+231. [6] PANTSULAIA G.R., On ordinary and standard products of infinite family of σ-finite measures and some of their applications.acta Math. Sin. (Engl. Ser.) 27 (2011), no. 3, [7] STEPHEN WILLARD, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts 10