Pointwise limits for sequences of orbital integrals
aa r X i v : . [ m a t h . D S ] F e b POINTWISE LIMITS FOR SEQUENCES OF ORBITALINTEGRALS
CLAIRE ANANTHARAMAN-DELAROCHE
Abstract.
In 1967, Ross and Str¨omberg published a theorem about pointwiselimits of orbital integrals for the left action of a locally compact group G onto( G, ρ ), where ρ is the right Haar measure. In this paper, we study the samekind of problem, but more generally for left actions of G onto any measuredspace ( X, µ ), which leaves the σ -finite measure µ relatively invariant, in thesense that sµ = ∆( s ) µ for every s ∈ G , where ∆ is the modular function of G . As a consequence, we also obtain a generalization of a theorem of Civin,relative to one-parameter groups of measure preserving transformations.The original motivation for the circle of questions treated here dates back toclassical problems concerning pointwise convergence of Riemann sums relativeto Lebesgue integrable functions. Introduction
The study of almost everywhere convergence of Riemann sums is an old problem,with many ramifications (see [RW06] for a recent survey). Let us consider theinterval [0 , T = R / (2 π Z ), and let ρ be the normalizedLebesgue measure on T . Let f : [0 , → C be a measurable function. For n ∈ N ∗ and x ∈ T , the corresponding Riemann sum of f is defined by R n f ( x ) = 1 n n − X j =0 f (cid:0) x + jn (cid:1) . When f is Riemann integrable, for any x ∈ T we havelim n R n ( f )( x ) = Z f ( t ) dρ ( t ) . (1.1)When f is Lebesgue integrable, it is easily seen that ( R n ( f )) converges in meanto R f dρ (see for instance [RW06, § Mathematics Subject Classification.
Primary 47A35; Secondary 37A15, 28D05, 28C10,26A42.
Key words and phrases.
Ergodic theory, transformation groups, Haar measures, Riemann andLebesgue integrals, lattices. to date back to the paper [Jes34] of Jessen. Jessen’s theorem states that if ( n k ) isa sequence of positive integers such that n k divides n k +1 for every k then, for any f ∈ L ( T ), we have, for almost every x ∈ T ,lim k R n k f ( x ) = Z f ( t ) dρ ( t ) . (1.2)Soon after, Marcinkiewicz and Zygmund [MZ37] on one hand, and Ursell [Urs37]independently, gave examples of functions f ∈ L ( T ) for which (1.1) fails almosteverywhere. For instance, given < δ < f : x ∈ ]0 ,
7→ | x | − δ , is such an example (see [Urs37, Rud64]). Later, Rudin [Rud64] was even able toprovide many examples of bounded measurable functions f (characteristic func-tions indeed) such that, for almost every x ∈ T , the sequence ( R n f ( x )) diverges.Moreover, Rudin’s paper highlighted deep connections between pointwise conver-gence of Riemann sums along a given subsequence ( n k ) of integers and arithmeticalproperties of the subsequence, a question now widely developed. The following dif-ferent important question has also been considered by many authors : under whichkind of regularity conditions on f does the associated sequence ( R n ( f )) of Riemannsums converges a.e. (see [RW06] for these questions and many related ones).In this paper, we deal with another sort of problem, namely we study possibleextensions of Jessen’s result to general locally compact groups and dynamicalsystems.Let us first come back to Jessen’s theorem and give another formulation of thisresult. Denote by G the group T and set G k = Z / ( n k Z ). Then ( G k ) is an increasingsequence of closed subgroups of G , whose union is dense in G . If ρ k is the Haarprobability measure on G k , Jessen’s result reads as follows : for every f ∈ L ( T ),lim k →∞ Z G k f ( t + x ) dρ k ( t ) = Z G f dρ a.e. Under this form, this theorem has been extended by Ross and Str¨omberg tolocally compact groups. An assumption about the behaviour of the modular func-tions of the subgroups is needed. It is automatically satisfied in the abelian case(see [RS67] and Corollary 3.12 below).More generally, we are interested in this paper by the following questions.
Let G y ( X, µ ) be an action of a locally compact group G on a measured space ( X, µ ) ,where µ is σ -finite, and let ( G n ) n ∈ N be an increasing sequence of closed subgroupsof G , with dense union : (i) find conditions on the action and on the (right) Haar measures ρ n of G n , ρ of G , so that for every f ∈ L ( X, µ ) and every n , t ∈ G n f ( tx ) is ρ n -integrable for almost every x ∈ X ; OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 3 (ii) if (i) holds, study the pointwise convergence of the sequence of orbital in-tegrals Z G n f ( tx ) dρ n ( t ) . (iii) identify the pointwise limit, in case it exists. A necessary condition for (i) to be satisfied is that, for every Borel subset B of X with µ ( B ) < + ∞ , for every n ∈ N , and for almost every x ∈ X , ρ n ( (cid:8) t ∈ G n ; x ∈ t − B (cid:9) ) = Z G n B ( tx ) dρ n ( t ) < + ∞ . When µ is finite, say a probability measure, condition (i) implies that the groups G n are compact. If we normalize their Haar measures by ρ n ( G n ) = 1, the reversedmartingale theorem implies that, for f ∈ L ( X, µ ), the sequence of orbital integralsconverges pointwise and in mean (see [Tem92, Cor. 3.5, page 219] and Proposi-tion 3.1 below). This fact is well-known. Moreover, the limit is the conditionalexpectation of f with respect to the σ -field of G -invariant Borel subsets of X . Ofcourse, Jessen’s theorem is a particular case.When µ is only σ -finite, we cannot use the reversed martingale theorem anylonger and we shall need other arguments. As already said, Ross and Str¨ombergstudied the case of the left action G y ( G, ρ ). They normalized the right Haarmeasures ρ n and ρ in such a way that for every f ∈ C c ( G ) (the space of continuousfunctions with compact support on G ), we have lim n ρ n ( f ) = ρ ( f ). This is alwayspossible, due to a result of Fell (see [Bou63, Chap. VIII, § n , the modularfunction of G n is the restriction of the modular function of G . We shall namethis property the modular condition (MC). Under these assumptions, Ross andStr¨omberg proved that for every f ∈ L ( G, ρ ), one has, for almost every x ∈ G ,lim n Z G n f ( tx ) dρ n ( t ) = Z G f ( tx ) dρ ( t ) = Z G f ( t ) dρ ( t ) . Later, Ross and Willis [RW97] provided an example showing that the modularcondition does not always hold. They also proved that the Ross-Str¨omberg theoremalways fails when the modular condition is not satisfied.Therefore, in our paper we shall always assume that the increasing sequence ( G n ) of closed subgroups of G has a dense union and satisfies the modular condition(MC) . We shall denote by ∆ the modular function of G , so that λ = ∆ ρ , where λ is the left Haar measure of G . We also assume that G acts on ( X, µ ) in such a way In this paper we adopt the following convention : each time we write an integral R f , either f is non-negative, or it is implicitely contained in the statement that f is integrable. CLAIRE ANANTHARAMAN-DELAROCHE that µ is ∆ -relatively invariant under the action , in the sense that sµ = ∆( s ) µ for s ∈ G . An important example is the left action G y ( G, ρ ).Under these assumptions, we give a necessary and sufficient condition for thepointwise limit theorem to be satisfied.
Theorem ( The two following properties are equivalent : (a) X is a countable union of Borel subsets B k of finite measure, such that forevery k and almost every x ∈ X , we have ρ ( (cid:8) t ∈ G ; x ∈ t − B k (cid:9) ) = Z G B k ( tx ) dρ ( t ) < + ∞ . (b) For every f ∈ L ( X, µ ) and for almost every x ∈ X , lim n Z G n f ( tx ) dρ n ( t ) = Z G f ( tx ) dρ ( t ) . A crucial intermediate step is the above mentioned Ross-Str¨omberg theorem.For completeness, we provide a proof of this result, partly based on one of theideas contained in [RS67].We give examples where our theorem 3.13 applies. In particular, as an easyconsequence we get our second main result :
Theorem ( Consider G y ( X, µ ) , where now the σ -finite measure µ isinvariant. Let ( G n ) n ∈ N be an increasing sequence of lattices in G , whose unionis dense. We fix a Borel fundamental domain D for G and we normalize ρ by ρ ( D ) = 1 . Then, for every G -invariant function f ∈ L ( X, µ ) and for almostevery x ∈ X , we have lim n | G n ∩ D | X t ∈ G n ∩ D f ( tx ) = Z D f ( tx ) dρ ( t ) , where | G n ∩ D | is the cardinal of G n ∩ D . This gives a simple way to extend a result of Civin [Civ55], which treated thecase G = R by a different method, apparently not directly adaptable to moregeneral locally compact groups G .2. Notation and conventions
In this paper, locally compact spaces are implicitely assumed to be Hausdorffand σ -compact. A measured space ( X, µ ) is a Borel standard space equipped witha (non-negative) σ -finite measure µ . Note that our assumptions imply the unimodularity of G . OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 5
Let G be a locally compact group. We denote by ∆ its modular function, andby λ and ρ respectively its left and right Haar measures, so that λ = ∆ ρ . Byan action of G on a measured space ( X, µ ) we mean a Borel map G × X → X ,( t, x ) tx , which is a left action and leaves µ quasi-invariant. In fact we shallneed the following stronger property : Definition 2.1.
Given an action G y ( X, µ ), we say that µ is ∆ -relatively invari-ant if sµ = ∆( s ) µ for all s ∈ G .Note that this property is satisfied for the left action G y ( G, ρ ).In all our statements, we shall consider an increasing sequence ( G n ) of closedsubgroups of G with dense union. Then ∆ n , ρ n , λ n = ∆ n ρ n will be the modularfunction and Haar measures of G n , respectively. We shall have to choose appro-priate normalizations of ρ and ρ n , n ∈ N . For a compact group, we usually chooseits Haar measure to have total mass one (but see remark 2.3 below).As already mentioned in the introduction, there is also a natural normalizationof the Haar measures as follows (see [Bou63, Chap. VIII, § Definition 2.2.
Let G be a locally compact group and ρ a right Haar measure on G . Let ( G n ) be an increasing sequence of closed subgroups, whose union is densein G . There is an essentially unique normalization of the right Haar measures ρ n of the G n such that, for every continuous function f on G with compact support,we have lim n ρ n ( f ) = ρ ( f ). In this case, we shall say that the sequence ( ρ n ) ofright Haar measures is normalized with respect to ρ . We shall also say that it is a Fell normalization . Remark . When G is compact, the Fell normalization is the classical normaliza-tion, where the Haar measures are probability measures. On the other hand, whenthe G n are compact whereas G is not, it is easily seen that the Fell normalizationimplies that lim n ρ n ( G n ) = + ∞ . For instance, let G be a countable discrete groupwhich is the union of an increasing sequence ( G n ) of finite subgroups ( e.g. thegroup S ∞ of finite permutations of the integers). Then, if ρ is the counting mea-sure on G , the normalization of the sequence ρ n with respect to ρ is the sequenceof counting measures, for which we have ρ n ( G n ) = | G n | , the cardinal of G n .Finally, another property of the sequence ( G n ) will be fundamental in this paper.It was already present in the work of Ross and Str¨omberg [RS67] and later shownto be crucial (see [RW97]). Definition 2.4.
Let G be a locally compact group and ( G n ) an increasing sequenceof closed subgroups of G . We say that ( G n ) satisfies the modular condition (MC) if,for every n , the modular function ∆ n of G n is the restriction to G n of the modularfunction ∆ of G . CLAIRE ANANTHARAMAN-DELAROCHE
Let us explain the interest of this condition. Given an action G y ( X, µ )leaving ∆-relatively invariant the measure µ , the following property of the actionholds : for every Borel function f : X × G → R ∗ + (or any µ ⊗ λ -integrable function f : X × G → C ), we have Z X × G f ( tx, t ) dµ ( x ) dρ ( t ) = Z X × G f ( x, t ) dµ ( x ) dλ ( t )= Z X × G f ( x, t − ) dµ ( x ) dρ ( t ) . (2.1)This property will be essential throughout this paper, and we shall need it toremain satisfied for all the restricted actions G n y ( X, µ ). This requires that therestriction of ∆ to G n is the modular function of G n .3. Limit theorems
Compacity assumptions.
In this section, the Haar measure of every com-pact group will have total mass one.
Proposition 3.1.
Let G be a locally compact group, acting in a measure preservingway on a probability space ( X, µ ) . Let ( G n ) be an increasing sequence of compactsubgroups of G , whose union is dense in G . Let f ∈ L ( X, µ ) . (a) For a.e. x ∈ X , we have lim n →∞ R G n f ( tx ) dρ n ( t ) = E ( f |I )( x ) , where E ( f |I ) is the conditional expectation of f with respect to the σ -field I of G -invariant Borel subsets of X . (b) If morever G is compact, then E ( f |I )( x ) = R G f ( tx ) dρ ( t ) for almost every x ∈ X .Proof. (a) Observe first that t ∈ G n f ( tx ) is ρ n -integrable for almost every x ,since Z X (cid:16) Z G n | f ( tx ) | dρ n ( t ) (cid:17) dµ ( x ) = Z X | f ( x ) | dµ ( x ) < + ∞ . We set R n ( f )( x ) = R G n f ( tx ) dρ n ( x ). Obviously, this function is G n -invariant andit is µ -integrable. Moreover, let A be a G n -invariant Borel subset of X . Then wehave Z A R n ( f )( x ) dµ ( x ) = Z X × G n A ( x ) f ( tx ) dµ ( x ) dρ n ( t )= Z X × G n A ( t − x ) f ( x ) dµ ( x ) dρ n ( t )= Z A f ( x ) dµ ( x ) , OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 7 since A is G n -invariant. Therefore R n ( f ) is the conditional expectation of f withrespect to the σ -field B n of Borel G n -invariant subsets of X . The sequence ( B n ) of σ -fields is decreasing. The reversed martingale theorem [Nev72, page 119] statesthat the sequence ( R n ( f )) converges µ -a.e. and in mean to the conditional ex-pectation of f with respect to the σ -field B ∞ = ∩ n B n of ( ∪ n G n )-invariant Borelsubsets of X . Since ∪ n G n is dense in G and µ is finite, B ∞ is also the σ -field of G -invariant Borel subsets of X .The proof of (b) is straightforward. (cid:3) Remark . Let µ be a σ -finite measure on a Borel space ( X, B ), and let ( B n )be a decreasing sequence of σ -fields. Assume that ( X, B n , µ ) is σ -finite for every n . For f ∈ L ( X, B , µ ), the conditional expectation E ( f |B n ) is well defined, as aRadon-Nikod´ym derivative. In [Jer59], Jerison has proved that lim n →∞ E ( f |B n )( x )exists almost everywhere. Moreover, if ( X, B ∞ , µ ) is σ -finite, the limit is E ( f |B ∞ ).Otherwise, one may write X as the disjoint union of two elements V, W of B ∞ ,where V is a countable union of elements of B ∞ of finite measure while any subset of W that belongs to B ∞ has measure 0 or ∞ . By [Jer59, § n →∞ E ( f |B n )( x ) =0 a.e. on W .This observation can be used to prove that proposition 3.1 still holds under theweaker assumption that µ is σ -finite. Indeed, it is enough to prove that when H is any compact group acting in measure preserving way on a σ -finite measuredspace ( X, B , µ ), then ( X, B H , µ ) is still σ -finite, where B H is the σ -field of Borel H -invariant subsets. To show this fact, consider a strictly positive function f ∈ L ( X, B , µ ). As seen in the proof of proposition 3.1, the function x R ( f )( x ) = R H f ( tx ) dρ ( t ) is H -invariant and µ -integrable and we deduce that ( X, B H , µ ) is σ -finite from the fact that R ( f ) is strictly positive everywhere. Finally, for every f ∈ L ( X, B , µ ), the conditional expectation E ( f |B H ) may be defined and we haveobviously R ( f )( x ) = E ( f |B H ).We shall give another proof of proposition 3.1 (b), for a compact group G anda σ -finite measure µ , in corollary 3.14.In the rest of the paper we are interested in the more general situation where µ is a σ -finite measure on X , and where the subgroups G n are not assumed to becompact. In particular G is not always unimodular.3.2. General case : local results.
Let G y ( X, µ ) be a measured G -space.When the measure µ is not finite, it may be useful to study the restriction ofthe action to every Borel subset B of X such that µ ( B ) < ∞ , even if B is not G -invariant . We extend to X every function defined on B , by giving it the value0 on X \ B . In particular, we have L ( B, µ ) ⊂ L ( X, µ ). The restriction of µ to the Borel subspace B will be denoted by the same letter. CLAIRE ANANTHARAMAN-DELAROCHE
Note that Borel functions of the form x ρ ( { t ∈ G ; tx ∈ B } ) = Z G B ( tx ) dρ ( t )are G -invariant. This crucial fact will be used repeatedly and without mention inthe sequel. Theorem 3.3.
Let ( G n ) be an increasing sequence of closed subgroups of a locallycompact group G , with dense union and satisfying condition (MC). Let G y ( X, µ ) be an action leaving the measure ∆ -relatively invariant. Let B be a Borel subsetof X with µ ( B ) < + ∞ . For a.e. x ∈ B and every n ∈ N , we assume that < ρ n ( { t ∈ G n ; tx ∈ B } ) < + ∞ , where ρ n is a right Haar measure on G n . Then,for every f ∈ L ( B, µ ) , the averaging sequence of functions x ∈ B ρ n ( { t ∈ G n ; tx ∈ B } ) Z G n f ( tx ) dρ n ( t ) converges, almost everywhere on B and in L norm, to an element of L ( B, µ ) .Proof. We first check that for every n and almost every x ∈ B , the function t ∈ G n f ( tx ) is ρ n -integrable. This is a consequence of the following computation(where we use equality (2.1)) : Z B ρ n ( { s ∈ G n ; sx ∈ B } ) (cid:16) Z G n | f ( tx ) | dρ n ( t ) (cid:17) dµ ( x )= Z X × G n B ( x ) | f ( tx ) | ρ n ( { s ∈ G n ; sx ∈ B } ) dµ ( x ) dρ n ( t )= Z X × G n B ( tx ) | f ( x ) | ρ n ( { s ∈ G n ; sx ∈ B } ) dµ ( x ) dρ n ( t )= Z B | f ( x ) | ρ n ( { s ∈ G n ; sx ∈ B } ) (cid:16) Z G n B ( tx ) dρ n ( t ) (cid:17) dµ ( x )= Z B | f ( x ) | dµ ( x ) < + ∞ . In addition, we see that x ∈ B R G n f ( tx ) dρ n ( t ) ρ n ( { s ∈ G n ; sx ∈ B } ) is µ -integrable on B .Denote by R n ( f ) this function defined on B .Let O n ( B ) be the equivalence relation on B induced by the G n -action : for x, y ∈ B , x ∼ O n ( B ) y if there exists t ∈ G n with x = ty . We denote by B n ( B )the σ -field of Borel subsets of B invariant under this equivalence relation. Observethat R n ( f ) is invariant under O n ( B ). It is also straightforward to check that R n ( f ) is the conditional expectation of f with respect to B n ( B ). Then, again theconclusion follows from the reversed martingale theorem. (cid:3) OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 9
Concerning the pointwise convergence of sequences of orbital integrals, we im-mediately get :
Corollary 3.4.
Under the assumptions of the previous theorem, the followingconditions are equivalent : (a) lim n →∞ ρ n ( { t ∈ G n ; tx ∈ B } ) exists a.e. on B ; (b) for every f ∈ L ( B, µ ) , lim n →∞ R G n f ( tx ) dρ n ( t ) exists a.e. on B . General case : global results.
In order to study the problem globally, weshall need the following lemma, insuring the integrability of orbital functions.
Lemma 3.5.
Let G y ( X, µ ) be an action leaving the measure ∆ -relatively in-variant. The two following conditions are equivalent : (i) X = ∪ B k , where every B k is a Borel subset of X , with µ ( B k ) < + ∞ , suchthat for almost every x ∈ X , ρ ( { t ∈ G ; tx ∈ B k } ) < ∞ . (ii) For every f ∈ L ( X, µ ) , f x : t f ( tx ) belongs to L ( G, ρ ) for a.e. x ∈ X .Proof. (ii) ⇒ (i) is obvious : write X as a countable union of Borel subsets B k offinite measure and take f = B k .(i) ⇒ (ii). Let f ∈ L ( X, µ ) + . It suffices to show that for every k and a.e. x ∈ B k , the function t f ( tx ) is ρ -integrable. We set A = (cid:26) x ∈ X ; Z G B k ( tx ) dρ ( t ) = 0 (cid:27) . Note that A is G -invariant.We first check that for a.e. x ∈ B k ∩ A , we have R G f ( tx ) dρ ( t ) = 0. Indeed, Z B k ∩ A Z G f ( tx ) dρ ( t ) dµ ( x ) = Z X × G B k ∩ A ( tx ) f ( x ) dρ ( t ) dµ ( x )= Z X f ( x ) (cid:16) Z G B k ∩ A ( tx ) dρ ( t ) (cid:17) dµ ( x ) = 0 . The first equality uses relation (2.1) and the last one follows from the observa-tion that B k ∩ A ( tx ) = 0 implies x ∈ A since A is G -invariant. Hence, we get R G f ( tx ) dρ ( t ) = 0 a.e. on B k ∩ A . Now let us consider Z B k \ A ρ ( { s ∈ G ; sx ∈ B k } ) (cid:16) Z G f ( tx ) dρ ( t ) (cid:17) dµ ( x ). This in-tegral is equal to Z X × G B k \ A ( x ) 1 ρ ( { s ∈ G ; sx ∈ B k } ) f ( tx ) dµ ( x ) dρ ( t )= Z X × G B k \ A ( tx ) 1 ρ ( { s ∈ G ; sx ∈ B k } ) f ( x ) dµ ( x ) dρ ( t )= Z X \ A f ( x ) (cid:16) Z G B k \ A ( tx ) ρ ( { s ∈ G ; sx ∈ B k } ) dρ ( t ) (cid:17) dµ ( x ) ≤ Z X \ A f ( x ) dµ ( x ) < + ∞ , since, for every x such that tx ∈ B k \ A we have x ∈ G ( B k \ A ) = GB k \ A , and Z G B k \ A ( tx ) dρ ( t ) ≤ Z G B k ( tx ) dρ ( t ) = ρ ( { s ∈ G ; sx ∈ B k } ) , with 0 < ρ ( { s ∈ G ; sx ∈ B k } ) < + ∞ a.e. on X \ A. It follows that R G f ( tx ) dρ ( t ) < + ∞ for amost every x ∈ B k \ A . (cid:3) Remark . The assumption of this lemma holds for instance when G acts on( X, µ ), where µ is a ∆-relatively invariant Radon measure on a locally compactspace X , the action being continuous with closed orbits and compact stabilizers.Indeed, in this situation, for x ∈ X , the natural map G/G x → Gx , where G x isthe stabilizer of x , is an homeomorphism. Then if B is an open relatively compactsubset of X , the set { t ∈ G ; tx ∈ B } is open and relatively compact in G andthe conclusion follows. Particular cases are proper actions, and more generallyintegrable actions [Rie04].We shall now give a condition sufficient to guarantee the pointwise convergenceof sequences of orbital integrals for every f ∈ L ( X, µ ). Note that the integrabilityof the functions appearing in the statement below follows from lemma 3.5.
Theorem 3.7.
Let ( G n ) be an increasing sequence of closed subgroups in G , withdense union and satisfying condition (MC). Let G y ( X, µ ) be an action leavingthe measure ∆ -relatively invariant. We assume that X = ∪ X k where ( X k ) is anincreasing sequence of Borel subspaces, such that for all k , (i) µ ( X k ) < + ∞ ; (ii) ρ n ( { t ∈ G n ; tx ∈ X k } ) > for almost every x ∈ X k and every n ; (iii) there exists c k > such that for almost every x ∈ X , sup n ρ n ( { t ∈ G n ; tx ∈ X k } ) ≤ c k . OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 11
The following conditions are equivalent : (a) for every k , lim n ρ n ( { t ∈ G n ; tx ∈ X k } ) exists for a.e. x ∈ X k ; (b) the pointwise limit lim n R G n f ( tx ) dρ n ( t ) exists a.e. on X , for every f ∈ L ( X, µ ) ; (c) there exists a dense subset D of L ( X, µ ) such that the pointwise limit lim n R G n f ( tx ) dρ n ( t ) exists a.e. on X , for every f in D .Remark . Assume that for every x ∈ X k and every n , ρ n ( { t ∈ G n ; tx ∈ X k } ) ≤ c k . (3.1)Then, assumption (iii) of the above theorem is fulfilled. Indeed, by invariance, if(3.1) holds for x ∈ X k , it also holds for x ∈ G n X k . On the other hand, if x / ∈ G n X k then { t ∈ G n ; tx ∈ X k } = ∅ , and therefore ρ n ( { t ∈ G n ; tx ∈ X k } ) = 0 ≤ c k .For the proof of theorem 3.7, we need the following lemma which repeats argu-ments from [RS67, Lemma 3]. Lemma 3.9.
Let ( G n ) be an increasing sequence of closed subgroups in G , withdense union and satisfying condition (MC). Let G y ( X, µ ) be an action leavingthe measure ∆ -relatively invariant. Let X k ⊂ X satisfying conditions (i) and (iii)of the previous theorem and let f : X → R + be a Borel function. For x ∈ X , set f ⋆ ( x ) = sup n Z G n f ( tx ) dρ n ( t ) , and for α > , set Q α = { x ∈ X ; f ⋆ ( x ) > α } . Then we have αµ ( Q α ∩ X k ) ≤ c k Z Q α f dµ. (3.2) Proof.
For n ∈ N , we set φ n ( x ) = R G n f ( tx ) dρ n ( t ) and we introduce the subsets E n = { x ∈ X ; φ n ( x ) > α } D n = ( x ∈ X ; sup ≤ l ≤ n φ l ( x ) > α ) . We fix an in integer N . It is enough to show that αµ ( D N ∩ X k ) ≤ c k Z D N f dµ since Q α is the increasing union of the sets D N , N ≥ We decompose D N as the union ∪ Nn =1 F n , where the subsets F n are mutuallydisjoint, and defined by F n = E n ∩ ( N [ l = n +1 E cl ) . Observe that G n E n = E n , and therefore G n F n = F n for every n . We have αµ ( F n ∩ X k ) ≤ Z F n ∩ X k φ n ( x ) dµ ( x ) ≤ Z X × G n F n ∩ X k ( x ) f ( tx ) dµ ( x ) dρ n ( t ) ≤ Z X × G n F n ∩ X k ( tx ) f ( x ) dµ ( x ) dρ n ( t ) ≤ Z X f ( x ) (cid:16) Z G n F n ∩ X k ( tx ) dρ n ( t ) (cid:17) dµ ( x ) . For F n ∩ X k ( tx ) to be non-zero, it is necessary that x ∈ G n ( X k ∩ F n ) = G n X k ∩ F n .It follows that αµ ( F n ∩ X k ) ≤ Z F n f ( x ) (cid:16) Z G n F n ∩ X k ( tx ) dρ n ( t ) (cid:17) dµ ( x ) ≤ c k Z F n f ( x ) dµ ( x ) . The conclusion is then an immediate consequence of the fact that D N is the unionof the mutually disjoint subsets F n , 1 ≤ n ≤ N . (cid:3) Remark . As a particular case, we shall use the following assertion. Let G y ( X, µ ) be a G -action such that µ is ∆-relatively invariant. Let X k be a Borelsubset of X with µ ( X k ) < + ∞ . Assume the existence of c k such that for almostevery x ∈ X , ρ ( { t ∈ G ; tx ∈ X k } ) ≤ c k . Let f : X → R + be a Borel function. For α >
0, set ˜ Q α = { x ∈ X ; R G f ( tx ) dρ ( t ) > α } . Then, we have αµ ( ˜ Q α ∩ X k ) ≤ c k Z ˜ Q α f dµ. Proof of theorem 3.7. (b) ⇒ (a) is obvious. Let us show that (a) ⇒ (c). Let f ∈ L ( X, µ ), null outside X k . Fix p > k . By theorem 3.3, we know thatlim n R G n f ( tx ) dρ n ( t ) ρ n ( { s ∈ G n ; sx ∈ X p } ) exists a.e. on X p . If (a) holds, we immediately get the existence of lim n R G n f ( tx ) dρ n ( t ) almost ev-erywhere on X p and therefore on the union X of the X p . Now, observe that suchfunctions f , supported in some X k , form a dense subspace of L ( X, µ ). OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 13
Finally, let us prove that (c) implies (b). We introduceΛ( f )( x ) = lim N → + ∞ (cid:16) sup n,m ≥ N | φ n ( f )( x ) − φ m ( f )( x ) | (cid:17) . We fix an integer p , and we shall show that Λ( f )( x ) = 0 for almost every x ∈ X p .This will end the proof. Take g ∈ D . We have Λ( g ) = 0 a.e. on X andΛ( f ) = Λ( f ) − Λ( g ) ≤ Λ( f − g ) ≤ | f − g | ⋆ . Given α >
0, it follows from lemma 3.9, that µ ( { x ∈ X p ; ∆( f )( x ) > α } ) ≤ µ ( { x ∈ X p ; | f − g | ⋆ > α/ } ) ≤ c p α k f − g k . Since we can choose g so that k f − g k is as close to 0 as we wish, we see that µ ( { x ∈ X p ; ∆( f )( x ) > α } ) = 0, from which we get µ ( { x ∈ X p ; ∆( f )( x ) > } ) =0. (cid:3) We apply theorem 3.7 to the following situation where, in addition, it is possibleto identify the limit.
Theorem 3.11.
Let G act properly on a locally compact σ -compact space X andlet µ be a ∆ -relatively invariant Radon measure on X . Let ( G n ) be an increasingsequence of closed subgroups, whose union is dense in G and which satisfies themodular condition (MC). We assume that the sequence ( ρ n ) of Haar measures isnormalized with respect to ρ . Then, for every f ∈ L ( X, µ ) , we have, for a.e. x ∈ X , lim n →∞ Z G n f ( tx ) dρ n ( t ) = Z G f ( tx ) dρ ( t ) . (3.3) Proof.
Let ( X k ) be an increasing sequence of open relatively compact subspacesof X , with X = ∪ X k . Of course, since the action is proper, we have0 < ρ n ( { t ∈ G n ; tx ∈ X k } ) < + ∞ for every x ∈ X k . Let us show that condition (iii) of theorem 3.7 is also fulfilled.Set K k = { t ∈ G ; tX k ∩ X k = ∅} . This set is relatively compact. We choose acontinuous function ϕ on G , with compact support, such that K k ≤ ϕ . We have ∀ n ∈ N , ∀ x ∈ X k , ρ n ( { t ∈ G n ; tx ∈ X k } ) ≤ ρ n ( ϕ ) . Since lim n ρ n ( ϕ ) = ρ ( ϕ ) < + ∞ , we obtain the existence of a constant c k such that ∀ n ∈ N , ∀ x ∈ X k , ρ n ( { t ∈ G n ; tx ∈ X k } ) ≤ c k . Now (iii) of theorem 3.7 is satisfied, by remark 3.8. Clearly, we may also choose c k such that, as well, ρ ( { t ∈ G ; tx ∈ X k } ) ≤ c k for all x ∈ X . The required integrability conditions for (3.3) follow from lemma 3.5. The exis-tence of the limit is an immediate consequence of theorem 3.7, applied to the space D = C c ( X ) of continuous functions with compact support in X . We use the factthat for every x ∈ X and f ∈ C c ( X ), the function t f ( tx ) is continuous withcompact support. Hence, by the normalization of the ρ n , we have the existence oflim n R G n f ( tx ) dρ n ( t ). Here, we even know that the limit is R G f ( tx ) dρ ( t ), for every x . It remains to identify the limit for every f ∈ L ( X, µ ). We set˜Λ( f )( x ) = lim N →∞ (cid:16) sup n ≥ N (cid:12)(cid:12)(cid:12)(cid:12)Z G n f ( tx ) dρ n ( t ) − Z G f ( tx ) dρ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:17) . As in the proof of theorem 3.7, we fix p , and we only need to show that ˜Λ( f )( x ) = 0for almost every x ∈ X p . Let g be a continuous function with compact support on X . We have˜Λ( f )( x ) = ˜Λ( f )( x ) − ˜Λ( g )( x ) ≤ ˜Λ( f − g )( x ) ≤ lim N (cid:16) sup n ≥ N (cid:12)(cid:12)(cid:12)(cid:12)Z G n ( f ( tx ) − g ( tx )) dρ n ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)Z G ( f ( tx ) − g ( tx )) dρ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | f − g | ⋆ + Z G | f ( tx ) − g ( tx ) | dρ ( t ) . Given α >
0, we have µ ( n x ∈ X p ; ˜Λ( f )( x ) > α o ) ≤ µ ( { x ∈ X p ; | f − g | ⋆ > α/ } )+ µ ( (cid:26) x ∈ X p ; Z G | f ( tx ) − g ( tx ) | dρ ( t ) > α/ (cid:27) ) ≤ c p α k f − g k . The last inequality follows from lemma 3.9 and remark 3.10. Now, we approximate f by a sequence ( f n ) of continuous functions with compact support. This gives µ ( n x ∈ X p ; ˜Λ( f )( x ) > α o ) = 0. The conclusion is obtained by letting α go to0. (cid:3) As a particular case, we obtain the following result of Ross and Str¨omberg. Incontrast to their proof, we do not use the theorem of Edwards and Hewitt ([EH65,Theorem 1.6]) on pointwise limits of sublinear operators whose ranges are familiesof measurable functions.
Corollary 3.12 ([RS67]) . Let G be a locally compact group, and ( G n ) be an in-creasing sequence of closed subgroups, whose union is dense in G and which satisfies OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 15 the modular condition (MC). We assume that the sequence ( ρ n ) of Haar measuresis normalized with respect to ρ . Then, for every f ∈ L ( G, ρ ) , we have lim n Z G n f ( tx ) dρ n ( t ) = Z G f ( t ) dρ ( t ) , a.e. Proof.
We apply theorem 3.11 to G y ( G, ρ ). (cid:3) We can now state our main theorem.
Theorem 3.13.
Let G y ( X, µ ) be an action on a measured space, leaving themeasure ∆ -relatively invariant. Let ( G n ) be an increasing sequence of closed sub-groups, whose union is dense in G and which satisfies the modular condition (MC).We assume that the sequence ( ρ n ) of Haar measures is normalized with respect to ρ . The two following properties are equivalent : (a) X is a countable union of Borel subsets B k of finite measure, such that forevery k and almost every x ∈ X , we have ρ ( { t ∈ G ; tx ∈ B k } ) = Z G B k ( tx ) dρ ( t ) < + ∞ . (3.4)(b) For every f ∈ L ( X, µ ) and for almost every x ∈ X , lim n Z G n f ( tx ) dρ n ( t ) = Z G f ( tx ) dρ ( t ) . Proof.
Assumption (b) contains the assertion that for every f ∈ L ( X, µ ) andalmost every x ∈ X , the function f x : t f ( tx ) is ρ -integrable. Thus, obviously(b) implies (a).Let us show that (a) implies (b). Let f ∈ L ( X, µ ). By lemma 3.5, there existsa conull subset E ⊂ X , such that for every x ∈ E , the function f x : t f ( tx ) isin L ( G, ρ ). We apply to f x the previous corollary. There exists a conull subset A x in G , such that for every s ∈ A x :(i) for n ∈ N , t ∈ G n f x ( ts ) in ρ n -integrable ;(ii) lim n R G n f x ( ts ) dρ n ( t ) = R G f x ( t ) dρ ( t ).Denote by D the set of all ( s, x ) ∈ G × X for which • t ∈ G n f ( tsx ) = f x ( ts ) is ρ n -integrable for all n , • t ∈ G f ( tsx ) = f x ( ts ) is ρ -integrable, • lim R G n f ( tsx ) dρ n ( t )) = R G f ( tsx ) dρ ( t ).Then D is a Borel subset of G × X . Moreover, D ⊃ { ( s, x ) ; x ∈ E, s ∈ A x } . It follows, by using twice the Fubini-Tonelli theorem, that D is conull, and thatfor almost every s ∈ G , we have, for almost x ∈ X :(1) f sx is ρ n -integrable for n ∈ N , and is ρ -integrable ;(2) lim n R G n f ( tsx ) dρ n ( t ) = R G f ( tx ) dρ ( t ).Choose such a s and let C ( s ) be a conull subset of X for which (1) and (2) occur.Then for any y ∈ sC ( s ), which is also conull, we have the required properties. (cid:3) Corollary 3.14.
Let G be a compact group acting on a measured space ( X, µ ) in such a way that the σ -finite measure µ is invariant. Let ( G n ) be an increasingsequence of closed subgroups of G , whose union is dense in G . We choose the Haarmeasures to have total mass . Then for every f ∈ L ( X, µ ) we have lim n → + ∞ Z G n f ( tx ) dρ n ( t ) = Z G f ( tx ) dρ ( t ) . Theorem 3.13 also applies to several other situations. We have already men-tioned in remark 3.6 the case of continuous actions with closed orbits and compactstabilizers. We now give another example of application.
Corollary 3.15.
Let G be a locally compact group acting on a measured space ( X, µ ) is such a way that the σ -finite measure µ is invariant. Let ( G n ) be anincreasing sequence of closed subgroups of G , whose union is dense in G and whichsatisfies the modular condition (MC). We assume that the sequence ( ρ n ) of Haarmeasures is normalized with respect to ρ . Let f ∈ L ( G × X, ρ ⊗ µ ) . Then for ρ ⊗ µ almost every ( s, x ) ∈ G × X , we have lim n →∞ Z G f ( ts, tx ) dρ n ( t ) = Z G f ( ts, tx ) dρ ( t ) . Proof.
We apply theorem 3.13 to B k = U k × V k , where ( U k ) k is a sequence ofrelatively compact open subsets of G with ∪ U k = G , and where ( V k ) k is a sequenceof Borel subsets of X , of finite measure, with ∪ V k = X . It suffices to observe that { t ∈ G ; t ( s, x ) ∈ U k × V k } ⊂ { t ∈ G ; ts ∈ U k } , and ρ ( { t ∈ G, t ( s, x ) ∈ B k } ) ≤ ρ ( { t ∈ G ; ts ∈ U k } ) < + ∞ . (cid:3) Corollary 3.16.
Let G y ( X, µ ) and ( G n ) be as in the previous corollary. Let h ∈ L ( X, µ ) and let E be a Borel subset of G with ρ ( E ) < ∞ . Then, for almostevery s ∈ G , we have lim n →∞ Z Es ∩ G n h ( tx ) dρ n ( t ) = Z Es h ( tx ) dρ ( t ) . OINTWISE LIMITS FOR SEQUENCES OF ORBITAL INTEGRALS 17
Proof.
We apply the previous corollary with f ( t, x ) = E ( t ) f ( x ). (cid:3) Example 3.17.
Let G = R , acting on R be left translations, and for n ∈ N ,take G n = Z / (2 n Z ). The Haar measure on G n is normalized by giving the weight1 / (2 n ) to each point and we take for µ the Lebesgue measure ρ on R , normalizedby ρ ([0 , E be a Borel subset of R such that ρ ( E ) < + ∞ . The previouscorollary gives that for every f ∈ L ( R , ρ ) and for almost every s ∈ R , we havelim n →∞ n X { k ; k/ n ∈ E + s } f (cid:0) k n + x (cid:1) = Z E + s f ( t + x ) dρ ( t ) a.e. Let us take E = Q for example. For every s irrational, the above equality holds(both sides are 0). On the other hand, for s ∈ Q , and f = [0 , , this equality isfalse for every x .3.4. An extension of a theorem of Civin.
We are now interested by the fol-lowing problem : let G y ( X, µ ) as before, that is, the σ -finite measure µ is∆-relatively invariant. We are given an increasing sequence of closed subgroupsof G , with dense union, and satisfying the modular condition. Let E be a Borelsubset of X with ρ ( E ) < ∞ , and let f ∈ L ( X, µ ). Find conditions under whichlim n Z G n ∩ E f ( tx ) dρ n ( t ) = Z E f ( tx ) dρ ( t )almost everywhere.In [Civ55], Civin has considered the particular case where G = R , G n = Z / (2 n Z )as in example 3.17. Let ( t, x ) t + x be a measure preserving action of R onto ameasured space ( X, µ ). Civin’s result states that for every f ∈ L ( X, µ ) such that f (1 + x ) = f ( x ) a.e. , then for almost every x ∈ X ,lim n →∞ Z G n ∩ [0 , f ( t + x ) dρ n ( t ) = lim n →∞ n n X k =1 f ( k n + x )= Z f ( t + x ) dρ ( t ) . More generally, we have :
Theorem 3.18.
Let ( G n ) n ∈ N be an increasing sequence of lattices of G (therefore G is unimodular), with dense union, and let D be a fundamental domain for G .Let G y ( X, µ ) be a measure preserving action. We assume that the Haar measureof G is normalized so that the volume of D is . Let f ∈ L ( X, µ ) be such that,for every t ∈ G , f ( tx ) = f ( x ) almost everywhere. Then lim n | G n ∩ D | X t ∈ G n ∩ D f ( tx ) = Z D f ( tx ) dρ ( t ) for a.e. x ∈ X .Proof. We normalize the Haar measure on G n by giving to each point the measure1 / | G n ∩ D | . This gives a normalized sequence of Haar measures with respect to ρ .Corollary 3.16 gives that, for almost every s ∈ G , we havelim n | G n ∩ Ds | X t ∈ G n ∩ Ds f ( tx ) = Z Ds f ( tx ) dρ ( t ) . For every t ∈ G n ∩ Ds , there exists a unique g ∈ G such that gt ∈ D . Due to the G -invariance of f , we have f ( tx ) = f ( gtx ) and therefore1 | G n ∩ Ds | X t ∈ G n ∩ Ds f ( tx ) = 1 | G n ∩ D | X t ∈ G n ∩ D f ( tx ) . On the other hand, by [Bou63, Corollaire, Page 69], the G -invariance of f impliesthat the integral R Ds f ( tx ) dρ ( t ) does not depend on the choice of the fundamentaldomain : we have R Ds f ( tx ) dρ ( t ) = R D f ( tx ) dρ ( t ). (cid:3) References [Bou63] N. Bourbaki. ´El´ements de math´ematique. Fascicule XXIX. Livre VI: Int´egration.Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et repr´esentations . Actualit´esScientifiques et Industrielles, No. 1306. Hermann, Paris, 1963.[Civ55] P. Civin. Abstract Riemann sums.
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