Definable convolution and idempotent Keisler measures
aa r X i v : . [ m a t h . L O ] A p r DEFINABLE CONVOLUTION AND IDEMPOTENT KEISLERMEASURES
ARTEM CHERNIKOV AND KYLE GANNON
Abstract.
We initiate a systematic study of the convolution operation onKeisler measures, generalizing the work of Newelski in the case of types. Adapt-ing results of Glicksberg, we show that the supports of generically stable (orjust definable, assuming NIP) measures are nice semigroups, and classify idem-potent measures in stable groups as invariant measures on type-definable sub-groups. We establish left-continuity of the convolution map in NIP theories,and use it to show that the convolution semigroup on finitely satisfiable mea-sures is isomorphic to a particular Ellis semigroup in this context. Introduction
Various notions and ideas from topological dynamics were introduced into themodel-theoretic study of definable group actions by Newelski [18, 19]. A funda-mental observation is that certain spaces of types over a definable group carry anatural algebraic structure of a (left-continuous) semigroup, with respect to the“independent product” of types. In a rather wide context, this operation can beextended from types to general
Keisler measures on a definable group (i.e. finitelyadditive probability measures on the Boolean algebra of definable subsets), where itcorresponds to convolution of measures. We first recall the classical setting. When G is a locally-compact topological group, then the space of regular Borel probabilitymeasures on G is equipped with the convolution product: if µ and ν are measureson G , then their product is the measure µ ∗ ν on G defined via µ ∗ ν ( A ) = Z y ∈ G Z x ∈ G χ A ( x · y ) dµ ( x ) dν ( y ) , for an arbitrary Borel set A ⊆ G (where χ A is the characteristic function of A ).And a measure µ is idempotent if µ ∗ µ = µ . A classical theorem of Wendel [28]shows that if G is a compact topological group and µ is a regular Borel probabilitymeasure on G , then µ is idempotent if and only if the support of µ is a compactsubgroup of G , and the restriction of µ to this subgroup is the (bi-invariant) Haarmeasure. Wendel’s result was extended to locally compact abelian groups by Rudin[24] and Cohen [5], and this line of research continued into the study of the structureof idempotent measures on (semi-)topological semigroups, in particular in the workof Glicksberg [12, 11] and Pym [22, 23].In this paper we consider the counterpart of these developments in the definablecategory, i.e. for definable groups and Keisler measures on them. In particular, weaim to address the following questions.(Q1) Under what conditions the convolution product of two global Keisler mea-sures can be defined?(Q2) What algebraic structures arise from idempotency of a Keisler measure? Date : May 28, 2019. (Q3) Is there a connection between the convolution semigroups of Keisler mea-sures and Ellis semigroups?We begin by reviewing some (mostly standard) material on Keisler measures inSection 2: we recall various classes of measures (invariant, Borel-definable, finitelysatisfiable, finitely approximable, smooth), summarize the relationship betweenthem (in general, as well as in NIP and stable theories) and discuss supports ofmeasures. In particular, in Proposition 2.11 we give a topological characterizationof the space of measures finitely satisfiable over a small model M , and in Lemma2.10 we make a couple of observations on invariantly supported measures (i.e. globalmeasures such that all types in their support are (automorphism-)invariant over afixed small model).In Section 3.1 we extend the usual product ⊗ of Borel-definable measures toa slightly larger context. Namely, when defining µ ⊗ ν , we only require the levelfunctions of the measure µ to be Borel restricted to the support of ν (Definition 3.1).It is equivalent to the standard definition when µ is Borel-definable, but allows toevaluate the product of an arbitrary invariant measure µ with an arbitrary type p for example (and this extends the usual independent product of invariant types,see Proposition 3.5). In relation to (Q1), in Section 3.2 we define the convolutionoperation on ∗ -Borel pairs of Keisler measures in terms of this generalized productof measures (Definition 3.8) and observe some of its basic properties, in particularthat it extends the independent product of arbitrary invariant types in a group(Proposition 3.11).In Section 3.3, we begin investigating idempotent Keisler measures. In Propo-sition 3.17 we observe that every invariant measure on a type-definable subgroupis idempotent (the extended ⊗ -product is needed for this to hold without any de-finability assumptions on the invariant measure). Mirroring the classical situationin Wendel’s theorem, the expectation is that in tame contexts all idempotent mea-sures should arise in this way. In the case of a definably amenable NIP group,invariant measures were classified in [3]. We observe in Proposition 3.18 that atype-definable subgroup of bounded index of a definably amenable NIP group isstill definably amenable (and the analysis from [3] extends to it). We also pointthat, as a consequence of Wendel’s theorem, idempotent measures finitely supportedon realized types correspond to finite subgroups (Proposition 3.20); and that in anabelian NIP group, the class of idempotent dfs (= definable and finitely satisfiable)measures is closed under convolution (Proposition 3.21).In Section 4, we study the supports of idempotent Keisler measures (question(Q2) above). In the proof of Wendel’s theorem (as well as Glicksberg’s proof in theabelian semitopological semigroup case [11]), an idempotent regular Borel measure µ is associated to a closed subgroup given by its support. In particular, S ( µ ) isa closed group and µ | S ( µ ) is its associated (bi-invariant) Haar measure. In thegeneral model-theoretic context the situation is not as nice (see Examples 4.1 and4.2). However, adapting some of Glicksberg’s work to our context, we show that if µ is definable, invariantly supported and idempotent, then ( S ( µ ) , ∗ ) (with respectto the usual independent product of invariant types) is a compact, left-continuoussemigroup with no closed two-sided ideals (Corollary 4.4 and Theorem 4.7). Thisassumption is satisfied when µ is a dfs measure in an arbitrary theory, or when µ is an arbitrary definable measure in an NIP theory. We also deduce that if sup( µ )has no proper closed left ideals, then µ is “generically” invariant restricting to its EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 3 support (Corollary 4.16). It follows that in abelian stable groups, the supportsof the idempotent measures are precisely the closed subgroups of the convolutionsemigroup on the space of types (Corollary 4.18); which leads to a quick descriptionof idempotent measures in strongly minimal groups (Example 4.19).In Section 5 we classify idempotent measures on a stable group, demonstratingthat they are precisely the invariant measures on its type-definable subgroups. Moreprecisely, every idempotent measure is the unique invariant Keisler measure on itsown (type-definable) stabilizer. Our proof relies on the results of the previoussection and a variant of Hrushovski’s group chunk theorem due to Newelski [17].Concerning question (Q3), it was observed by Newelski [18] that the convolutionsemigroup ( S x ( G , G ) , ∗ ) on the space of global types finitely satisfiable in a smallmodel G ≺ G is isomorphic to the enveloping Ellis semigroup E ( S x ( G , G ) , G ) of theaction of G on this space of types. Ellis semigroups for definable group actions inthe context of NIP theories were previously considered in e.g. [2, 3], to which werefer for a general discussion. In Section 6, under the NIP assumption, we obtainan analogous description for the convolution semigroup ( M x ( G , G ) , ∗ ) on the spaceof global Keisler measures finitely satisfiable in a small model. Namely, in Theorem6.10 we show that it is isomorphic to the Ellis semigroup E ( M x ( G , G ) , conv( G )) ofthe action of conv( G ), the convex hull of G in the space of global measures finitelysatisfiable on G , on this space of measures (see Remark 6.11 on why the convexhull is necessary). Our proof relies in particular on left-continuity of convolution ofinvariant measures in NIP theories established in Section 6.2 using approximationarguments with smooth measures. Acknowledgements.
This work constitutes part of the Ph.D dissertation of thesecond named author. We thank Sergei Starchenko for several helpful conversationson the topics of this paper. Both authors were partially supported by the NSFCAREER grant DMS-1651321, and Gannon was additionally supported by theNSF conference grant DMS-1922826.2.
Preliminaries on Keisler measures
Basic facts about Keisler measures.
For the majority of this article, wefocus on global Keisler measures and their relationship to small elementary sub-models. In this section we recall some of the material from [16, 13, 14, 15, 4, 9],and refer to e.g. [25, Chapter 7] for a more detailed introduction to the subject, or[26, 1] for a survey.Given r , r ∈ R and ε ∈ R > , we write r ≈ ε r if | r − r | < ε . Let T bea first order theory in a language L and assume that U is a sufficiently saturatedmodel of T (we make no assumption on T unless explicitly stated otherwise). Inthis section, we write x, y, z, . . . to denote arbitrary finite tuples of variables. If x is a tuple of variables and A ⊆ U , then L x ( A ) is the collection of formulas withfree variables in x and parameters from A , up to logical equivalence (which weidentify with the corresponding Boolean algebra of definable subsets of U x ). Wewrite L x for L x ( ∅ ). Given a partitioned formula ϕ ( x ; y ), we let ϕ ∗ ( y ; x ) := ϕ ( x ; y )be the partitioned formula with the roles of x and y reversed. As usual, S x ( A )denotes the space of types over A , and if A ⊆ B ⊆ U then S x ( B, A ) (respectively, S inv x ( B, A )) denotes the closed set of types in S x ( B ) that are finitely satisfiable in A (respectively, invariant over A ). For any set A ⊆ U , a Keisler measure over A in variables x is a finitely additive probability measure on L x ( A ). We denote A. CHERNIKOV AND K. GANNON the space of Keisler measures over A (in variables x ) as M x ( A ). Every element of M x ( A ) is in unique correspondence with a regular Borel probability measure on thespace S x ( A ), and we will routinely use this correspondence. If M (cid:22) M (cid:22) U aresmall models, then there is an obvious restriction map r from M x ( M ) to M x ( M )and we denote r ( µ ) simply as µ | M . Conversely, every µ ∈ M x ( M ) admits anextension to some µ ′ ∈ M x ( M ) (not necessarily a unique one).The space M x ( A ) is a compact Hausdorff space with the topology induced from[0 , L x ( A ) . This is the coarsest topology on the set M x ( A ) such that for anycontinuous function f : S x ( A ) → R , the map µ → R f dµ is continuous. If M (cid:22) M ,then under this topology, the restriction map r is continuous. We identify every p ∈ S x ( A ) with the corresponding Dirac measure δ p ∈ M x ( A ), and under thisidentification S x ( A ) is a closed subset of M x ( A ).We recall several important properties of global measures that will make anappearance in this article. Definition 2.1.
Let µ ∈ M x ( U ) be a global Keisler measure.(1) µ is invariant if there is a small model M ≺ U such that for any partitioned L ( M )-formula ϕ ( x ; y ) and any b, b ′ ∈ U y , if b ≡ M b ′ then µ ( ϕ ( x ; b )) = µ ( ϕ ( x ; b ′ )). In this case, we say µ is M -invariant . We let M inv x ( U , M )denote the closed set of all M -invariant measures in M x ( U ).(2) Assume that µ is M -invariant and ϕ ( x ; y ) is a partitioned L ( M )-formula.We define the map F ϕµ,M : S y ( M ) → [0 ,
1] by F ϕµ,M ( q ) = µ ( ϕ ( x ; b )), where b | = q (this is well-defined by M -invariance of µ ).We will often write F ϕµ instead of F ϕµ,M when the base model M is clearfrom the context.(3) µ is Borel-definable (respectively, definable ) if there is M ≺ U such that µ is M -invariant and for any partitioned L ( M )-formula ϕ ( x ; y ), the map F ϕµ,M is Borel-measurable (respectively, continuous). In this case, we saythat µ is Borel-definable over M (respectively, definable over M ).(4) µ is finitely satisfiable in M ≺ U if for any L x ( U )-formula ϕ ( x ), if µ ( ϕ ( x )) > U | = ϕ ( a ) for some a ∈ M x . We let M x ( U , M ) denote the closed setof measures in M x ( U ) which are finitely satisfiable in M .(5) µ is dfs if there is M ≺ U such that µ is both definable over M and finitelysatisfiable in M . Similarly, if this is the case, we say that µ is dfs over M .(6) Given a ∈ ( U x ) <ω , with a = ( a , ..., a n ), the associated average measure Av a ∈ M x ( U ) is defined byAv a ( ϕ ( x )) := |{ i : U | = ϕ ( a i ) }| n for any ϕ ( x ) ∈ L x ( U ).(7) µ is finitely approximated if there is M ≺ U such that for any L ( M )-formula ϕ ( x ; y ) and any ε ∈ R > , there exist n ∈ N ≥ and ¯ a ∈ ( M x ) n such thatfor any b ∈ U y , µ ( ϕ ( x ; b )) ≈ ε Av ¯ a ( ϕ ( x ; b )). In this case, we call ¯ a a ( ϕ, ε ) -approximation for µ , and we say µ is finitely approximated in M .(8) µ is smooth if there exists a small model M ≺ U such that for any N with M (cid:22) N (cid:22) U , there exists a unique measure µ ′ ∈ M x ( N ) such that µ ′ | M = µ | M . In this case, we say that µ is smooth over M .These properties are related as follows. EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 5
Fact 2.2. (1) In any theory T , given µ ∈ M x ( U ) , over any given M ≺ U :(a) µ is smooth ⇒ µ is finitely approximated [15, Corollary 2.6] ;(b) µ is finitely approximated ⇒ µ is dfs (e.g. by Fact 2.3 below);(c) µ is definable ⇒ µ is Borel-definable;(d) if µ is either Borel-definable or finitely satisfiable, then µ is invariant.(2) Assuming T is NIP, given µ ∈ M x ( U ) , over any M ≺ U we have addition-ally:(a) µ is invariant ⇒ µ is Borel-definable ( [14, Corollary 4.9] , or [25,Proposition 7.19] );(b) µ is dfs ⇒ µ is finitely approximated [15, Theorem 3.2] .(3) Assuming T is stable, given any µ ∈ M x ( U ) we have moreover:(a) µ is finitely approximated (see e.g. [4, Lemma 4.3] for a direct proof );(b) for every L -formula ϕ ( x ; y ) , there exist types ( p i ) i ∈ ω in S x ( U ) and ( r i ) i ∈ ω , r i ∈ [0 , such that P r i = 1 , and taking µ ′ := P r i · p i wehave µ ( ϕ ( x ; b )) = µ ′ ( ϕ ( x ; b )) for all b ∈ U y [16, Lemma 1.7] ;(c) If T is ω -stable, then there exist ( p i ) i ∈ ω in S x ( U ) and ( r i ) i ∈ ω , r i ∈ [0 , such that P r i = 1 and µ = P r i · p i (same as the proof of [16, Lemma1.7] , using boundedness of the global rank). We have the following characterization of definability (see e.g. [9, Proposition4.4]).
Fact 2.3.
The following are equivalent for µ ∈ M x ( U ) and M (cid:22) U .(1) The measure µ is definable over M .(2) For any partitioned L ( M ) -formula ϕ ( x ; y ) and any ε > , there existformulas Φ ( y ) , ..., Φ n ( y ) such that each Φ i ( y ) ∈ L y ( M ) , the collection { Φ i ( U ) : i ≤ n } forms a partition of U y , and if | = Φ i ( c ) ∧ Φ i ( c ′ ) , then | µ ( ϕ ( x, c )) − µ ( ϕ ( x, c ′ )) | < ε .(3) For every partitioned formula ϕ ( x ; y ) ∈ L ( M ) and every n ∈ N ≥ thereexist some L y ( M ) -formulas Φ ϕ, n i ( y ) with i ∈ I n := { , n , n , . . . , n − n , } such that:(a) the collection { Φ ϕ, n i ( U ) : i ∈ I n } forms a covering of U y (but notnecessarily a partition);(b) For every i ∈ I n and b ∈ U y , if U | = Φ ϕ, n i ( b ) then | µ ( ϕ ( x, b )) − i | < n . This easily implies the following.
Fact 2.4. If M (cid:22) N ≺ U and µ ∈ M x ( N ) is definable over M , then there existsa unique extension µ ′ ∈ M x ( U ) of µ which is definable over M (given by the samedefinition schema Φ as in Fact 2.3, and denoted µ | U ). In an NIP theory, every measure over a small model can be extended to a smoothmeasure over a slightly larger elementary extension ([16, Theorem 3.16], or [25,Proposition 7.9]).
Fact 2.5.
Let T be an NIP theory. Let M ≺ U and µ ∈ M x ( M ) . Then µ admits asmooth extension. I.e., there exist some ν ∈ M x ( U ) and some small M (cid:22) N ≺ U such that ν is smooth over N and ν | M = µ . Definition 2.6.
Given a Keisler measure µ ∈ M x ( A ), the support of µ is S ( µ ) = { p ∈ S x ( A ) : µ ( ϕ ( x )) > ϕ ( x ) ∈ p } . A. CHERNIKOV AND K. GANNON
Types in S ( µ ) are sometimes called weakly random with respect to µ in the litera-ture.We recall some properties of supports, with proofs for the sake of completeness. Proposition 2.7.
Let µ ∈ M x ( A ) .(1) Then for any ϕ ( x ) ∈ L x ( A ) such that µ ( ϕ ( x )) > , there exists some q ∈ S ( µ ) such that ϕ ( x ) ∈ q . In particular, S ( µ ) = ∅ .(2) S ( µ ) is a closed subset of S x ( A ) and µ ( S ( µ )) = 1 (and S ( µ ) is the smallestset of types under inclusion with this property).Proof. (1) Without loss of generality, ϕ ( x ) ≡ x = x . Otherwise, we reiterate theproof with the normalization of µ to the definable set ϕ ( x ), i.e. considering theKeisler measure µ ′ defined by µ ′ ( ψ ( x )) := µ ( ψ ( x ) ∧ ϕ ( x )) µ ( ϕ ( x )) for all ψ ( x ). Assume that S ( µ ) = ∅ , then for every type p ∈ S x ( A ), there exists some ϕ p ( x ) ∈ p such that µ ( ϕ p ( x )) = 0. Then, µ ( ¬ ϕ p ( x )) = 1 for every p ∈ S x ( A ), hence for any n and p , ..., p n ∈ S x ( A ), we have T ni =1 ¬ ϕ p i ( x ) = ∅ . Then K = T p ∈ S x ( A ) ¬ ϕ p ( x ) = ∅ by compactness of S x ( A ). By if q ∈ K , then in particular ¬ ϕ q ( x ) ∈ q — acontradiction.(2) Assume that p S ( µ ). Then, there exists a formula ϕ p ( x ) such that ϕ p ( x ) ∈ p and µ ( ϕ p ( x )) = 0. Then, S x ( A ) \ S ( µ ) = [ p S ( µ ) ϕ p ( x ) . Therefore, S ( µ ) is closed. Assume that µ ( S x ( M ) \ S ( µ )) >
0. By regularity of µ ,there exists a clopen C ⊆ S x ( A ) \ S ( µ ) with positive measure. But by (1) we musthave C ∩ S ( µ ) = ∅ , a contradiction. (cid:3) Proposition 2.8.
Let A ⊆ B ⊆ U and µ ∈ M x ( B ) be arbitrary. Let r : S x ( B ) → S x ( A ) , q q | A be the restriction map. Then:(1) r ( S ( µ )) = S ( µ | A ) ;(2) the measure µ | A is the pushforward of µ along r , i.e. r ∗ ( µ ) = µ | A .Proof. (1) The map r is a continuous surjection between compact Hausdorff spaces.By Proposition 2.7(2), r ( S ( µ )) is compact (hence, closed), as the continuous imageof a compact set. Clearly r ( S ( µ )) ⊆ S ( µ | A ), and as r ( S ( µ )) is closed it suffices toshow that r ( S ( µ )) is a dense subset of S ( µ | A ). Indeed, assume that ϕ ( x ) ∈ L x ( A )and ϕ ( x ) ∩ S ( µ | A ) = ∅ . Then µ | A ( ϕ ( x )) >
0, hence µ ( ϕ ( x )) >
0, and by Proposition2.7(1) there exists some q ∈ S ( µ ) with ϕ ( x ) ∈ q . Hence ϕ ( x ) ∈ r ( q ), and so r ( S ( µ )) ∩ ϕ ( x ) = ∅ . And (2) is clear. (cid:3) Definition 2.9.
We say that µ ∈ M x ( U ) is invariantly supported if there existssome small model M ≺ U such that every type p ∈ S ( µ ) is M -invariant. Lemma 2.10.
Let µ ∈ M x ( U ) .(1) If µ is finitely satisfiable, then µ is invariantly supported.(2) If µ is invariantly supported, then µ is invariant.(3) If T is NIP, the µ is invariant if and only if it is invariantly supported.(4) In some theory, there exist a definable measure µ ∈ M x ( U ) and p ∈ S ( µ ) such that p is not invariant (over any small set). EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 7
Proof. (1) Clearly if µ is finitely satisfiable over M ≺ U , then every p ∈ S ( µ ) isalso finitely satisfiable in M .(2) Let M ≺ U be a small model such that every p ∈ S ( µ ) is invariant over M . If µ is not invariant over M , then there exist some ϕ ( x, y ) ∈ L xy and some b ≡ M b ′ in U y such that µ ( ϕ ( x, b )) = µ ( ϕ ( x, b ′ )). But then µ ( ϕ ( x, b ) △ ϕ ( x, b ′ )) > ϕ ( x, b ) △ ϕ ( x, b ′ ) ∈ p for some p ∈ S ( µ ) by Proposition 2.7 — contradicting M -invariance of p .(3) ( ⇒ ) holds by (2). For ( ⇐ ), we note that if µ is invariant over M ≺ U , thenevery global type p ∈ S ( µ ) doesn’t divide over M (given ϕ ( x, b ) ∈ p and an M -indiscernible sequence ( b i ) i ∈ ω in U y such that b i ≡ M b , we have that µ ( ϕ ( x, b i )) = µ ( ϕ ( x, b )) =: ε > i ; but then µ ( V i Let µ ∈ M x ( U ) and M ≺ U a small model. Then µ is finitelysatisfiable in M if and only if µ is in the closure of conv( M x ) (viewed as a subsetof M x ( U )) .Proof. Assume µ is finitely satisfiable in M . Let U be a basic open subset of M x ( U )containing µ . Say U = n \ i =1 { µ ∈ M x ( U ) : r i < µ ( ϕ i ( x )) < s i } for some n ∈ N , ϕ ( x ) , ..., ϕ n ( x ) ∈ L x ( U ) and r i , ..., r n , s , ..., s n ∈ [0 , { ϕ ( x ) , ..., ϕ n ( x ) } generates a finite Boolean subalgebra of L x ( U ). Let θ ( x ) , ..., θ m ( x ) be its atoms, and let Θ := { θ j ( x ) : µ ( θ j ( x )) > } . As µ is finitelysatisfiable in M , for each θ j ( x ) ∈ Θ, there exists some a j ∈ M x such that | = θ j ( a j ).Let ν := P θ j ∈ Θ µ ( θ j ( x )) δ a j ∈ M x ( U ). Then we have µ ( ϕ i ( x )) = ν ( ϕ i ( x )) forall 1 ≤ i ≤ n (note that a j | = θ i ⇐⇒ i = j ), so ν ∈ U ∩ conv( M x ). Hence µ ∈ cl(conv( M x )).Conversely, suppose µ ∈ cl(conv( M x )) and let ψ ( x ) ∈ L x ( U ) be such that µ ( ψ ( x )) > 0. Consider the open set U ψ := { ν ∈ M x ( U ) : 0 < ν ( ϕ ( x )) } containing µ . Since µ is in the closure of conv( M x ), there exists some µ ψ = P ni =1 r i δ a i , where a i ∈ M x for all i and µ ψ ∈ U ψ . But then U | = ψ ( a i ) for at least one i . (cid:3) Definable convolution and idempotent measures A. CHERNIKOV AND K. GANNON Extended product of measures. We begin by defining a slight general-ization of the product of measures that encompasses both the usual independentproduct of Borel-definable measures and the standard Morley product of invarianttypes (without any definability assumptions), and also allows to take products of G -invariant measures in arbitrary theories. This is accomplished by slightly tweakingthe domain of the integral in the usual definition of the ⊗ -product. Definition 3.1. Let µ ∈ M x ( U ), ν ∈ M y ( U ), and ϕ ( x, y, c ) ∈ L xy ( U ). We saythat the triple ( µ, ν, ϕ ) is Borel if there exists some N ≺ U such that:(1) c ⊆ N ;(2) for any q ∈ S ( ν | N ) and d, d ′ ∈ U y with d, d ′ | = q , we have that µ ( ϕ ( x, d, c )) = µ ( ϕ ( x, d ′ , c ));(3) the map F ϕµ,N : S ( ν | N ) → [0 , 1] is Borel, where F ϕµ,N ( q ) = µ ( ϕ ( x, d, c )) forsome/any d | = q .We say that the ordered pair ( µ, ν ) is Borel if ( µ, ν, ϕ ) is Borel for any ϕ ( x, y, c ) ∈L xy ( U ). Definition 3.2. Assume that ( µ, ν ) is Borel, and let N be any small elementarysubmodel of U witnessing this (as in Definition 3.1). Then we define the productmeasure µ ˜ ⊗ ν ∈ M xy ( U ) as follows: µ ˜ ⊗ ν ( ϕ ( x, y, c )) = Z S ( ν | N ) F ϕµ,N dν N , with the notation from Definition 3.1, where ν N is the restriction of the regularBorel measure ν | N to the compact set S ( ν | N ).We need to check that ˜ ⊗ is well-defined. Proposition 3.3. Assume that ( µ, ν, ϕ ) is Borel. Then, µ ˜ ⊗ ν ( ϕ ( x ; y, c )) does notdepend on the choice of N (as in Definition 3.1).Proof. This proof is essentially the same as for ⊗ (see e.g. [25, Proposition 7.19]).Assume that ( µ, ν, ϕ ) is Borel with respect to both M and N . We may assume that M ⊆ N (taking a common extension). By Proposition 2.8, let r : S ( ν N ) → S ( ν M )be the restriction map; then F ϕµ,M ◦ r = F ϕµ,N and the pushforward of the measure ν N , namely r ∗ ( ν N ), is equal to ν M . Hence we have: Z S ( ν | M ) F ϕµ,M d ( ν M ) = Z S ( ν | M ) F ϕµ,M dr ∗ ( ν N ) = Z S ( ν | N ) (cid:16) F ϕµ,M ◦ r (cid:17) dν N = Z S ( ν | N ) F ϕµ,N dν N . (cid:3) We recall the independent product of invariant types (see e.g. [25, Section 2.2]). Fact 3.4. (1) Assume M ≺ U is a small submodel, p ∈ S inv x ( U , M ) and U ′ ≻U . There there exists a unique type p ′ ∈ S inv x ( U ′ , M ) extending p . Then forany A ⊆ U ′ , we write p | A to denote p ′ | A .(2) Assume that p ∈ S x ( U ) , q ∈ S y ( U ) and p is invariant. Then p ⊗ q := tp ( a, b/ U ) ∈ S xy ( U ) for some/any b | = q and a | = p | U b (in some U ′ ≻ U ;this is well-defined by (1)). EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 9 (3) If p, q ∈ S x ( U , M ) (respectively, p, q ∈ S inv x ( U , M ) ), then p ⊗ q ∈ S xy ( U , M ) (respectively, p ⊗ q ∈ S inv xy ( U , M ) ). The product ˜ ⊗ extends both the independent product on invariant types andthe product of Borel definable probability measures in arbitrary theories. Proposition 3.5. (1) Let µ ∈ M x ( U ) and ν ∈ M y ( U ) . Assume that µ isBorel-definable. Then, µ ⊗ ν = µ ˜ ⊗ ν .(2) If µ ∈ M x ( U ) is invariant and q ∈ S x ( U ) is arbitrary, then ( µ, δ q ) is Boreland µ ˜ ⊗ δ q is well-defined.(3) Let p ∈ S x ( U ) and q ∈ S y ( U ) , and p is invariant. Then δ p ⊗ q = δ p ˜ ⊗ δ q ,where p ⊗ q ) is the free product (see Fact 3.4).Proof. (1) It is easy to see that R S y ( N ) F ϕµ d ( ν | N ) = R S ( ν | N ) F ϕµ dν N as long as theintegral on the left hand side is well-defined — which is the case when µ is Boreldefinable.(2) Let ϕ ( x, y ) ∈ L ( U ), and let N ≺ U containing all the parameters from ϕ besuch that µ is invariant over N . Note that the map F ϕµ : S y ( N ) → [0 , 1] need not beBorel definable. However S ( δ q | N ) is a single point as q is a type, hence F ϕµ ↾ S ( δ q | N ) is trivially Borel.(3) By (2), ( δ p , δ q ) is Borel. Let N ≺ U witness this, and let b ∈ U y , b | = q | N .Then δ p ˜ ⊗ δ q ( ϕ ( x ; y )) = Z sup( δ q | N ) F ϕδ p d ( δ q ) N = F ϕδ p ( q | N ) = ( ϕ ( x, b ) ∈ p, ¬ ϕ ( x, b ) ∈ p. That is, δ p ˜ ⊗ δ q ( ϕ ( x ; y )) = 1 if and only if ϕ ( x, y ) ∈ tp( a, b/N ) for some/any b | = q | N and a | = q | Nb . (cid:3) From now on we will simply write ⊗ instead of ˜ ⊗ to denote this extended oper-ation when there is no ambiguity involved. Definition 3.6. We say that µ, ν ∈ M x ( U ) ( ⊗ -) commute if both ( µ, ν ) and ( ν, µ )are Borel, and µ ⊗ ν = ν ⊗ µ .We recall some facts about commuting measures. Fact 3.7. Assume that µ ∈ M x ( U ) , ν ∈ M y ( U ) and M ≺ U .(1) [15, Theorem 2.5] Assume that µ is Borel-definable over M and ν is smoothover M . Then, for any ϕ ( x ; y ) ∈ L xy ( M ) , we have that, Z S x ( M ) F ϕµ d ( ν | M ) = Z S y ( M ) F ϕ ∗ ν d ( µ | M ) . In particular, µ ⊗ ν = ν ⊗ µ .(2) ( [4, Proposition 2.13] or [6, Proposition 2.10] ) If µ and ν are finitely ap-proximated over M , then µ ⊗ ν = ν ⊗ µ . Definable convolution. Throughout this section, we let T be a first order L -theory expanding a group. We let G be a sufficiently saturated model of T , and G denotes a small elementary submodel. We use letters x, y to denote singleton variables, i.e. of the sort on which the group is defined. For any formula ϕ ( x, c ) ∈L x ( G ), we let ϕ ′ ( x, y, c ) = ϕ ( x · y, c ). Definition 3.8. Let µ, ν ∈ M x ( G ), and let ν y denote the measure in M y ( G ) suchthat for any ϕ ( y ) ∈ L y ( G ), ν y ( ϕ ( y )) = ν ( ϕ ( x )).(1) We say that ( µ, ν ) is ∗ -Borel if for every formula ϕ ( x, c ) ∈ L x ( G ), the triple( µ, ν y , ϕ ′ ) is Borel. We say that µ is ∗ -Borel if the pair ( µ, µ ) is ∗ -Borel.(2) If ( µ, ν ) is ∗ -Borel, then we define the (definable) convolution product of µ with ν as follows: µ ∗ ν ( ϕ ( x, c )) = µ ˜ ⊗ ν y ( ϕ ′ ( x, y, c )) = Z S ( ν y | G ) F ϕ ′ µ dν G ( y ) , where G is some/any small submodel of G witnessing that ( µ, ν y , ϕ ′ ) is Boreland ν G ( y ) is the Borel measure ν y restricted to S ( ν y | G ) (as in Definition3.2). We will routinely write this product simply as R F ϕ ′ µ dν when there isno possibility of confusion.Note that we are integrating over translates with respect to the right action of G , and in general throughout the article, when speaking about G -invariance andrelated notion, we will typically consider the action of G on the right . This choice ismade to make sure that this definition correctly extends Newelski’s product of types(Proposition 3.11), but of course all of our results hold with respect to left actionsmodulo obvious modifications. First we check that the convolution operation indeeddefines a measure. Fact 3.9. Let µ, ν ∈ M x ( G ) . If ( µ, ν ) is ∗ -Borel, then µ ∗ ν is a Keisler measure.Proof. Clearly µ ∗ ν ( x = x ) = 1 and µ ∗ ν ( ¬ ϕ ( x )) = 1 − µ ∗ ν ( ϕ ( x )). Assume that ψ ( x ) ∧ ψ ( x ) = 0. Let θ ( x ; y ) = ψ ( x · y ) ∨ ψ ( x · y ), and let G ≺ G contain all ofthe parameters. Then for any q ∈ S ( ν | G ) and b | = q we have F θµ ( q ) = µ ( θ ( x ; b )) = µ ( ψ ( x · b ) ∨ ψ ( x · b )). As ψ ( x ) ∧ ψ ( x ) = ∅ implies ψ ( x · b ) ∧ ψ ( x · b ) = ∅ , wehave F θµ ( q ) = µ ( ψ ( x · b )) + µ ( ψ ( x · b )) = F ψ ′ µ ( q ) + F ψ ′ µ ( q ) . Then ( µ ∗ ν )( ψ ( x ) ∨ ψ ( x )) = Z S ( ν | G ) F θµ dν G = Z S ( ν | G ) (cid:16) F ψ ′ µ + F ψ ′ µ (cid:17) dν G = Z S ( ν | G ) F ψ ′ µ dν G + Z S ( ν | G ) F ψ ′ µ dν G = ( µ ∗ ν )( ψ ( x )) + ( µ ∗ ν )( ψ ( x )) . (cid:3) This notion of convolution extends the notion of the product of invariant typesextensively studied by Newelski [18, 19] and others from the point of view of topo-logical dynamics. The following is easy using Fact 3.4. Fact 3.10. Let G ≺ G be a small model. Given p, q ∈ S inv x ( G , G ) , we define p ∗ q := tp( a · b/ G ) ∈ S inv x ( G , G ) , for some/any ( a, b ) | = p ⊗ q in a larger monstermodel. Then (cid:0) S inv x ( G , G ) , ∗ (cid:1) is a semigroup, with multiplication continuous in theleft coordinate: for each q ∈ S inv x ( G , G ) , the map − ∗ q : S inv x ( G , G ) → S inv x ( G , G ) iscontinuous. And ( S x ( G , G ) , ∗ ) is a closed sub-semigroup. Proposition 3.11. Let δ : S x ( G , G ) → M x ( G , G ) be the map δ ( p ) = δ p . Then δ isa topological embedding, and δ p ∗ q = δ p ∗ δ q for any p, q ∈ S x ( G , G ) . EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 11 Proof. Clearly δ is a topological embedding. Now let ϕ ( x ) ∈ L x ( G ) be arbitrary,then by Proposition 3.5(3) we have δ p ∗ q ( ϕ ( x )) = δ p x ⊗ q y ( ϕ ( x · y )) = δ p x ˜ ⊗ δ q y ( ϕ ( x · y )) = δ p ∗ δ q ( ϕ ( x )) . (cid:3) The next lemma follows by straightforward computations. Proposition 3.12. Let µ, µ , . . . , µ n , ν , . . . , ν m ∈ M x ( G ) be arbitrary, and assumethat the pairs ( µ i , ν j ) are ∗ -Borel for all ≤ i ≤ n, ≤ j ≤ m . Let a, a , . . . , a n ∈ G and r , . . . , r n , s , . . . , s m ∈ R ≥ be such that P ni =1 r i = P mj =1 s j = 1 . Then:(1) µ ∗ δ e = δ e ∗ µ = µ ,(2) δ a ∗ δ b = δ ab ,(3) ( δ a ∗ µ )( ϕ ( x )) = µ ( ϕ ( a · x )) for any ϕ ( x ) ∈ L x ( U ) ,(4) ( P ni =1 r i · µ i ) ∗ (cid:16)P mj =1 s j · ν j (cid:17) = P n,mi,j =1 r i · s j · ( µ i ∗ ν j ) ,(5) (( P ni =1 r i · δ a i ) ∗ µ ) ( ϕ ( x )) = P ni =1 r i · µ ( ϕ ( a i · x )) for any ϕ ( x ) ∈ L x ( U ) . Finally, we observe that the following properties of measures are preserved underconvolution. Proposition 3.13. Let µ, ν ∈ M x ( G ) be such that ( µ, ν ) is ∗ -Borel, and G ≺ G .(1) If µ, ν are definable over G , then µ ∗ ν is definable over G .(2) If µ, ν are finitely satisfiable over G , then µ ∗ ν is finitely satisfiable over G .(3) If µ, ν are finitely approximated over G , then µ ∗ ν is finitely approximatedover G .(4) If µ ( x = b ) = 0 for every b in G , then µ ∗ ν ( x = b ) = 0 for every b ∈ G .Proof. Claims (1), (2) and (3) are slight variations on the preservation of the corre-sponding properties with respect to ⊗ (see e.g. [15, Lemma 1.6] or [6, Proposition2.6] for (1) and (2), and [4, Proposition 2.13] or [6, Proposition 2.10] for (3)).(4) Note that µ ∗ ν ( x = b ) = µ ˜ ⊗ ν y ( x · y = b ) = Z S ( ν | G ) F ϕ ′ µ dν G . And F ϕ ′ µ ( q ) = µ ( x · c = b ) for some/any c | = q . Then, µ ( x · c = b ) = µ ( x = bc − ) = 0by assumption. Therefore, R F ϕ ′ µ dν G = R dν G = 0. (cid:3) Idempotent measures. We continue working in a theory expanding a group,and begin with some standard definitions. Definition 3.14. Let µ ∈ M x ( G ).(1) We say that µ is idempotent if µ is ∗ -Borel and µ ∗ µ = µ .(2) We say that µ is right-invariant if for any formula ϕ ( x ) ∈ L x ( G ) and any a ∈ G , we have µ ( ϕ ( x )) = µ ( ϕ ( x · a )). Definition 3.15. Let H be a type-definable subgroup of G , where H ( x ) is thepartial type defining the domain of H (which we associate with the closed setof types implying H ). Then H is definably amenable if there exists a measure µ ∈ M x ( G ) such that µ ( H ( x )) = 1, and for any ϕ ( x ) ∈ L x ( G ) and a ∈ H we have µ ( ϕ ( x )) = µ ( ϕ ( x · a )). In this case, we call µ right H -invariant . Remark 3.16. For NIP groups, existence of a right-invariant measure on H isequivalent to the existence of a left invariant measure on H (as well as a bi-invariantmeasure, see [3, Lemma 6.2]). Proposition 3.17. Let H be a type-definable, definably amenable subgroup of G ,defined by a partial type H ( x ) . Suppose that µ ∈ M x ( G ) is right H -invariant. Then µ is idempotent. Moreover, if ν is another measure such that ν ( H ( x )) = 1 , then ( µ, ν ) is ∗ -Borel and µ ∗ ν = µ .Proof. We show that for any measure ν ∈ M x ( G ) such that ν ( H ( x )) = 1, ( µ, ν )is ∗ -Borel and µ ∗ ν = µ . For ease of notation, we will identify ν with ν y . Fix aformula ϕ ( x ) ∈ L x ( G ). Let G be a small elementary submodel of G containing theparameters of H ( x ) and ϕ ( x ). Fix some q ∈ sup( ν | G ) ⊆ S y ( G ), then q ⊢ H ( y ).If not, then q ∈ S y ( G ) \ H ( y ). Since H ( y ) is closed, S y ( G ) \ H ( y ) is open, hence S y ( G ) \ H ( y ) = S i ∈ I ψ i ( y ) for some index set I and ψ i ∈ L y ( G ). Then ψ i ( y ) ∈ q for some i and since q ∈ S ( ν | G ), we know that ν ( ψ i ( y )) > 0. But this is acontradiction since ν ( H ( y )) = 1 and ψ i ( y ) is disjoint from H ( y ). Therefore, if b ∈ G and b | = q , then b ∈ H . Now, the function F ϕ ′ µ,G is constant on S ( ν | G ) since F ϕ ′ µ,G ( q ) = µ ( ϕ ( x · b )) = µ ( ϕ ( x )) by right H -invariance of µ , hence ( µ, ν ) is ∗ -Borel.And µ ∗ ν = µ as µ ∗ ν ( ϕ ( x )) = Z sup( ν | G ) F ϕ ′ µ,G dν G = Z sup( ν | G ) µ ( ϕ ( x )) dν G = µ ( ϕ ( x )) . In particular, ( µ, µ ) is Borel and µ ∗ µ = µ . (cid:3) The expectation is that in tame situations, all idempotent measures are of thisform for some type-definable subgroup. We will show that this is indeed the casewhen G is a stable group in Section 5, but for now we discuss some examples inwhich idempotent measures arise.If G is a definably amenable group, and H is a type-definable subgroup of finiteindex (hence definable), then H is definably amenable (if µ is a right-invariantmeasure on G , then µ H ( ϕ ( x )) := [ G : H ] · µ ( ϕ ( x ) ∩ H ( x )) gives a right-invariantmeasure on H ). This generalizes to subgroups of bounded index when G is NIP. Proposition 3.18. Assume that G is definably amenable and NIP, and let H be atype-definable subgroup of G of bounded index. Then H is also definably amenable. Proposition 3.18 follows from a slightly generalized construction of the G -invariantmeasures µ p from [13, 14, 3] in NIP groups. We will use some properties of the abso-lute type-definable connected component G , the intersection of all type-definablesubgroups of G of bounded index, and refer to the aforementioned texts for furtherdetails. To be compatible with our set up for convolutions, we work with G actingon the right . Let G ≺ G be a small model such that G is type-definable over G . Asusual, π : G → G / G is the surjective group homomorphism with π ( g ) dependingonly on tp( g/G ). Then G / G is a compact Hausdorff topological group with respectto the logic topology, i.e. a subset X of G / G is closed iff π − ( X ) is type-definable.The induced map S x ( G ) → G / G is continuous. With respect to this topology,closed subgroups of G / G are in a bijective correspondence with type-definablesubgroups of G of bounded index (equivalently, containing G ). Namely, if K isa closed subgroup of G / G , then H := π − ( K ) is a type-definable set containing G = π − ( e K ), and is a group since π is a group homomorphism (and vice versa). EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 13 Also, if H ⊆ G is type-definable, then K := π ( H ) ⊆ G / G is a closed subgroup (as π : S x ( G ) → G / G is a closed map).Recall that a global type p ∈ S x ( G ) is strongly f -generic over G if p · g is G -invariant for every g ∈ G . If G is definably amenable and G is an arbitrary smallmodel, there there exists a type strongly f -generic over G (see [14]). Moreover, asevery right translate of a strong f -generic over G is again a strong f -generic over G and G is a normal subgroup of G , we can always find one with p ( x ) ⊢ G ( x ). Proof of Proposition 3.18. Let K := π ( H ), then π − ( K ) = H (by the fourth iso-morphism theorem for groups), hence K is a closed subgroup of G / G . Denote by ν the right-invariant Haar measure on Borel subsets of K normalized by ν ( K ) = 1.Let p ∈ S x ( G ) be a strong f -generic over G , with p ⊢ G , so in particular p ⊢ H and p is G -invariant. For a formula ϕ ( x ) ∈ L x ( G ), let A ϕ,p := { ¯ g ∈ K : ϕ ( x ) ∈ p · ¯ g } . Then A ϕ,p is a Borel subset of K (as A ϕ,p = K ∩ (cid:8) ¯ g ∈ G / G : ϕ ( x ) ∈ p · ¯ g (cid:9) , andthe latter set is Borel by [14, Proposition 5.6]). We define µ p,ν ( ϕ ( x )) := ν ( A ϕ,p ) . Then we have the following. • µ p,ν is a Keisler measure with µ p,ν ( H ) = 1.It is easy to check that µ p,ν is a measure. And by regularity, µ p,ν ( H ) =inf { µ p,ν ( ψ ( x )) : H ( x ) ⊢ ψ ( x ) , ψ ( x ) ∈ L x ( G ) } , and as p ⊢ H ⊢ ψ for all such ψ , we have that A ψ,p = K by definition, hence µ p,ν ( ψ ) = 1). • µ p,ν is right H -invariant (as µ p,ν ( ϕ ( x ) · g ) = ν ( A ϕ · g,p ) = ν ( A ϕ,p · π ( g )) = ν ( A ϕ,p ) = µ p,ν ( ϕ ( x )) by K -invariance of ν , as π ( g ) ∈ K ).Hence H is definably amenable, witnessed by µ p,ν . (cid:3) Question 3.19. Is Proposition 3.18 true without the NIP assumption?Classification of measures supported on finite subsets of G follows from Wendel’stheorem. Proposition 3.20. If µ is a measure on G whose support is a finite collection ofrealized types, then µ is idempotent if and only if µ = |H| P a ∈H δ a for some finitesubgroup H of G .Proof. ( ⇐ ) is by Proposition 3.17.( ⇒ ) Assume that S ( µ ) = { a , ..., a n } = A ⊆ G . As µ is idempotent, S ( µ ) isclosed under multiplication (if not, then there exists c ∈ G \ A such that c = a i · a j and c for some i, j ; then µ ( x = c ) = 0, but µ ∗ µ ( x = c ) > A is closedunder products. As any finite subset of a group closed under products is a subgroup, A is a compact group, and µ | A is an idempotent measure on A . Therefore, by [28,Theorem 1], µ | A is the unique Haar measure on the subgroup S ( µ | A ) of A . But as S ( µ ) = A , we conclude that µ = n P a ∈ A δ a . (cid:3) Finally, we observe a sufficient condition for idempotence to be preserved underconvolution. Proposition 3.21. (1) Assume that G is abelian, µ, ν ∈ M x ( G ) are idempo-tent, and µ, ν ⊗ -commute. Then µ ∗ ν is idempotent. (2) In particular, if G is NIP and abelian, and both µ, ν are idempotent anddfs, then µ ∗ ν is idempotent and dfs.Proof. Fix a formula ϕ ( x ) ∈ L ( G ) and assume that G ≺ G witnesses that both( µ, ν, ϕ ) and ( ν, µ, ϕ ) are Borel. Then µ ∗ ν ( ϕ ( x )) = µ x ⊗ ν y ( ϕ ( x · y )) = ν y ⊗ µ x ( ϕ ( x · y )) . By change of variables and abelianity of G , we can conclude= ν x ⊗ µ y ( ϕ ( y · x )) = ν x ⊗ µ y ( ϕ ( x · y )) = ν ∗ µ ( ϕ ( x )) . Now, let λ = µ ∗ ν . Using associativity of ⊗ , λ ∗ λ = µ ∗ ν ∗ µ ∗ ν = µ ∗ µ ∗ ν ∗ ν = µ ∗ ν = λ. (2) follows from (1), Facts 2.2(2b) and 3.7, and Proposition 3.13. (cid:3) Supports of idempotent measures In this section, we will show that if µ is definable, invariantly supported (seeDefinition 2.9) and idempotent, then ( S ( µ ) , ∗ ) is a compact, left-continuous semi-group with no closed two-sided ideals. This assumption is satisfied when µ is adfs measure in an arbitrary theory (by Fact 2.10(1)), and when µ is an arbitrarydefinable measure in an NIP theory (by Fact 2.10(3)).We begin by considering two examples, which illustrate in particular that thesupport of an idempotent dfs Keisler measure need not be a group in general. Example 4.1. Let T = T doag be the complete theory of a divisible ordered abeliangroup in the language { + , <, , } . Let G be a monster model of T and consider G := Q as an elementary substructure in the natural way. Let p ∞ be the uniqueglobal type finitely satisfiable in G and extending { x > a : a ∈ Q } . Let p −∞ bethe unique global type finitely satisfiable in G and extending { x < a : a ∈ Q } .Let µ := δ p −∞ + δp ∞ , we claim that µ, δ p ∞ , δ p −∞ ∈ M x ( G ) are idempotent. ByProposition 3.13, the product δ α ∗ δ β for α, β ∈ { p ∞ , p −∞ } is finitely satisfiable in Q . Then, using Proposition 3.12, it is not hard to verify the following calculation: µ ∗ µ = (cid:16) δ p −∞ + 12 δ p ∞ (cid:17) ∗ (cid:16) δ p −∞ + 12 δ p ∞ (cid:17) = 14 (cid:16) p −∞ ∗ p −∞ (cid:17) + 14 (cid:16) p −∞ ∗ p ∞ (cid:17) + 14 (cid:16) p ∞ ∗ p −∞ (cid:17) + 14 (cid:16) p ∞ ∗ p ∞ (cid:17) = 14 p −∞ + 14 p ∞ + 14 p −∞ + 14 p ∞ = 12 p −∞ + 12 p ∞ = µ. We observe that while ( S ( δ p ∞ ) , ∗ ) and ( S ( δ p −∞ ) , ∗ ) are groups (with a single ele-ment), ( S ( µ ) , ∗ ) is not a group since it does not contain an identity element. Example 4.2. Let G = ( S , · , C ( x, y, z )) be the standard circle group over R , with C the cyclic clockwise ordering. Let T O be the corresponding theory. Let µ be theKeisler measure on this structure which corresponds to the restriction of the Haarmeasure on S . Let G be a monster model of T O such that S ≺ G . Then µ is smoothover S and admits a unique global extension ˜ µ . We remark that ˜ µ is right invariant,hence ˜ µ idempotent (Proposition 3.17). Let st : S x ( G ) → S be the standard partmap. Assume that p ∈ S (˜ µ ) and st( p ) = a . Then ϕ ε ( x ) := C ( a − ε, x, a + ε ) / ∈ p for every infinitesimal ε ∈ G ( x = a ∈ p as µ ( x = a ) = 0, and if ϕ ε ( x ) ∈ p ,then ˜ µ ( ϕ ε ( x ) ∧ x = a ) > 0, but ϕ ε ( G ) = { a } — contradicting finite satisfiability EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 15 of ˜ µ in G ). As the types are determined by the cuts in the circular order, itfollows that for every a ∈ S there are exactly two types a + ( x ) , a − ( x ) ∈ S (˜ µ )determined by whether C ( a + ε, x, b ) holds for every infinitesimal ε and b ∈ G , or C ( b, x, a − ε ) holds for every infinitesimal ε and b ∈ G , respectively. It follows that(sup(˜ µ ) , ∗ ) ∼ = S × { + , −} with multiplication defined by: a δ ∗ b γ = ( a · b ) δ for all a, b ∈ S and δ, γ ∈ { + , −} . Again, ( S ( µ ) , ∗ ) is not a group.Next we establish various properties of ( S ( µ ) , ∗ ) when µ is a global idempotentmeasure which is definable and invariantly supported. Given S , S ⊆ S x ( G ), wewrite S ∗ S := { p ∗ p ∈ S x ( G ) : p i ∈ S i } (under the assumption that all suchproducts are defined, i.e. assuming ( p , p ) is Borel for all p i ∈ S i ). The assumptionof being invariantly supported in the lemmas below is only needed to ensure that S ( µ ) ∗ S ( µ ) is defined (Fact 3.10). Proposition 4.3. Let µ, ν ∈ M x ( G ) . Assume that µ is definable, and both µ and ν are invariantly supported. Then:(1) S ( µ ) ∗ S ( ν ) ⊆ S ( µ ∗ ν ) ;(2) S ( µ ) ∗ S ( ν ) is a dense subset of S ( µ ∗ ν ) .Proof. (1) Assume that p ∈ S ( µ ) , q ∈ S ( ν ), and let ϕ ( x ) ∈ p ∗ q . Choose G ≺ G such that µ is definable over G , p, q are finitely satisfiable in G , and G contains allthe parameters from ϕ . We need to show that µ ∗ ν ( ϕ ( x )) > 0. Now, µ ∗ ν ( ϕ ( x )) = Z S ( ν | G ) F ϕ ′ µ,G dν G Since µ is definable, the map F ϕ ′ µ,G : S ( ν | G ) → [0 , 1] is continuous. Therefore, itsuffices to find some r ∈ S ( ν | G ) such that F ϕ ′ µ,G ( r ) > 0. Consider r := q | G . Then, F ϕ ′ µ,G ( q | G ) = µ ( ϕ ( x · b )), where b | = q | G . Then, ϕ ( x · b ) ∈ p and since p ∈ S ( µ ), wehave that µ ( ϕ ( x · b )) > 0. Hence, F ϕ ′ µ ( q | G ) > µ ∗ ν ( ϕ ( x )) > S ( µ ) ∗ S ( ν ) ⊆ S ( µ ∗ ν ). Fix some r ∈ S ( µ ∗ ν )and a formula ϕ ( x ) ∈ r . Assume that ϕ ( x ) ∈ r . We need to find p ∈ S ( µ ) and q ∈ S ( ν ) such that ϕ ( x ) ∈ p ∗ q . Choose G such that µ is definable over G , all typesin S ( µ ) , S ( ν ) are invariant over G , and G contains the parameters of ϕ ( x ). Since ϕ ( x ) ∈ r and r is in the support of µ ∗ ν , we know that µ ∗ ν ( ϕ ( x )) > 0. Therefore, R S ( ν | G ) F ϕ ′ µ,G d ( ν G ) > 0, and so there exists some t ∈ S ( ν | G ) such that F ϕ ′ µ,G ( t ) > c | = t , then µ ( ϕ ( x · c )) > 0. So, by Proposition 2.7(1), there exists p ∈ S ( µ ) suchthat ϕ ( x · c ) ∈ p . By Proposition 2.8, we let q ∈ S ( ν ) be such that q | G = t . Byconstruction, we then observe that ϕ ( x ) ∈ p ∗ q . (cid:3) Corollary 4.4. Assume that µ is definable, invariantly supported and idempotent.Then ( S ( µ ) , ∗ ) is a compact Hausdorff (with the subspace topology) semigroup whichis left-continuous, i.e. the map − ∗ q : S ( µ ) → S ( µ ) is continuous for each q ∈ S ( µ ) .Proof. By Proposition 2.7(2), S ( µ ) is a compact Hausdorff space. By Proposition4.3, S ( µ ) ∗ S ( µ ) ⊆ S ( µ ∗ µ ) = S ( µ ). Now, choose some G ≺ G such that µ isdefinable over G , and all types in S ( µ ) are invariant over G . Then ( S ( µ ) , ∗ ) is asub-semigroup of ( S inv x ( G , G ) , ∗ ) and ∗ is left-continuous by Fact 3.10. (cid:3) We now define some global functions which mimic the map y R f ( x · y ) dµ ( x ). Definition 4.5. Let µ ∈ M x ( G ) be definable, and fix ϕ ( x ) ∈ L x ( G ). We thendefine the global function D ϕ ′ µ : S y ( G ) → [0 , 1] via p µ ( ϕ ( x · c )), for some/any c | = p | G and G ≺ G small and containing the parameters of ϕ ( x ).Note that for any formula ϕ ( x ) ∈ L x ( G ), the map D ϕ ′ µ is continuous: D ϕ ′ µ = F ϕ ′ µ,G ◦ r , where r : S y ( G ) → S y ( G ) is the restriction map, and F ϕ ′ µ,G is continuousby definability of µ . The next two results are adapted from Glicksberg’s work onsemi-topological semigroups into the general model theory context. In particular,see [12, 11]. Proposition 4.6. Let µ ∈ M x ( G ) be definable, invariantly supported and idem-potent, and ϕ ( x ) ∈ L x ( G ) arbitrary. Assume that D ϕ ′ µ | S ( µ ) attains a maximum at q ∈ S ( µ ) (exists as this is a continuous function on a compact set). Then for any p ∈ S ( µ ) , we have that D ϕ ′ µ ( q ) = D ϕ ′ µ ( p ∗ q ) .Proof. Fix a small model G ≺ G such that µ is definable over G , and G containsthe parameters of ϕ ( x ). Let b | = q | G and let θ ( x ; y ) := ϕ (( x · y ) · b ). Now fixa larger submodel G ≺ G such that G b ⊂ G . Let δ := µ ( ϕ ( x · b )). Observethat then for any t ∈ S ( µ | G ), a | = t , and ˜ t ∈ S ( µ ) such that ˜ t | G = t , we have F θµ,G ( t ) = µ ( ϕ ( x · a ) · b ) = µ ( ϕ ( x · ( a · b ))) = D ϕ ′ µ (˜ t ∗ q ) ≤ D ϕ ′ µ ( q ) = δ (by theassumption on q ). We conclude that for any t ∈ S ( µ | G ), F θµ,G ( t ) ≤ δ . On the otherhand, δ = D ϕ ′ µ ( q ) = µ ( ϕ ( x · b )) = µ ∗ µ ( ϕ ( x · b )) = µ x ˜ ⊗ µ y ( θ ( x ; y ))= Z S ( µ | G ) F θµ,G dµ G . Therefore, F θµ = δ almost everywhere (with respect to µ G ). Since both maps arecontinuous, they are equal on S ( µ | G ). Finally, for any p ∈ S ( µ ) and a | = p , wehave: D ϕ ′ µ ( q ) = δ = F θµ,G ( p | G ) = µ ( ϕ (( x · a ) · b )) = µ ( ϕ ( x · ( a · b ))) = D ϕ ′ µ ( p ∗ q ) , as wanted. (cid:3) Theorem 4.7. Let µ ∈ M x ( G ) be definable, invariantly supported and idempotent.Let I ⊂ S ( µ ) be a closed two-sided ideal. Then, I = S ( µ ) .Proof. If I is dense in S ( µ ), then I = S ( µ ). So we may assume that I is not densein S ( µ ). Therefore, there exists some ϕ ( x ) ∈ L x ( G ) such that ϕ ( x ) ∩ S ( µ ) = ∅ and ϕ ( x ) ∩ I = ∅ . Let G ≺ G contain the parameters of ϕ , and such that µ is definableand invariantly supported over G . Claim 4.8. There exists some q ∈ S ( µ ) such that D ϕ ′ µ ( q ) > .Proof. Assume not. Let p, q ∈ S ( µ ) be arbitrary. Let b | = q | G , a | = p | Gb . Then µ ( ϕ ( x · b )) = D ϕ ′ µ ( q ) = 0 by assumption, hence | = ¬ ϕ ( a · b ) as p ∈ S ( µ ), so ϕ ( x ) / ∈ p ∗ q .Consider now the continuous characteristic function χ ϕ : S ( µ ) → { , } . ByProposition 4.3(2) and the previous paragraph, χ ϕ vanishes on a dense subset S ( µ ) ∗ S ( µ ) of S ( µ ), hence χ ϕ vanishes on S ( µ ). But this contradicts the choice of ϕ . (cid:3) EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 17 So there exists some q ∈ S ( µ ) such that D ϕ ′ µ ( q ) > 0. Then, since D ϕ ′ µ is contin-uous, it attains a maximum δ > r ∈ S ( µ ). Claim 4.9. For any h ∈ I , we have D ϕ ′ µ ( h ) = 0 .Proof. Let h ∈ I . Then D ϕ ′ µ ( h ) = µ ( ϕ ( x · b )), where b | = h | G . Then µ ( ϕ ( x · b )) = µ ( { p ∈ S ( µ ) : ϕ ( x · b ) ∈ p } ) = µ ( { p ∈ S ( µ ) : ϕ ( x ) ∈ p ∗ h } ) . As I is a left ideal, we have S ( µ ) ∗ h ⊆ I . By assumption, ϕ ( x ) ∩ I = ∅ , and so wehave { p ∈ S ( µ ) : ϕ ( x ) ∈ p ∗ h } = ∅ . Therefore, D ϕ ′ µ ( h ) = 0. (cid:3) Finally, since I is a right ideal, we have that h ∗ r ∈ I . Therefore, using Propo-sition 4.6 and the claim, 0 < D ϕ ′ µ ( r ) = D ϕ ′ µ ( h ∗ r ) = 0 . Therefore, we obtain a contradiction. (cid:3) Corollary 4.10. Assume that | S ( µ ) | > , i.e. µ is not a type. Then S ( µ ) containsno zero elements, i.e. there is no element p ∈ S ( µ ) such that for any q in S ( µ ) , p ∗ q = q ∗ p = p .Proof. If p is a zero-element, then { p } is a closed two sided ideal. (cid:3) We make some further observations on the structure of the semigroup sup( µ )under the additional assumptions on the idempotent measure µ . We recall thefollowing structural theorem of Ellis (with the roles of multiplication on the leftand on the right exchanged everywhere). Fact 4.11. [7, Proposition 1.12] Assume that ( S, · ) is a compact Hausdorff semi-group which is left-continuous (i.e. such that for any a ∈ S , the map − · a : S → S is continuous). Then, there exists a minimal left ideal I (which is automaticallyclosed). We let J ( I ) = { i ∈ I : i = i } be the set of idempotents in I .(1) J ( I ) is non-empty.(2) For every p ∈ I and i ∈ J ( I ) , we have that p · i = p .(3) I = S { i · I : i ∈ J ( I ) } , where the union is over disjoint sets, and each set i · I is a group with identity i .(4) I · q is a minimal right ideal for all q ∈ S . Assume that µ ∈ M x ( G ) is definable, invariantly supported and idempotent. Then( S ( µ ) , ∗ ) is a semigroup satisfying the assumption of Fact 4.11 by Corollary 4.4. Definition 4.12. We let I µ denote the minimal (closed) left ideal of ( S ( µ ) , ∗ ) (itexits by Fact 4.11). We say that µ is minimal if I µ = S ( µ ).In particular, if µ is minimal, then S ( µ ) is a disjoint union of subgroups. Example 4.13. For example, the measure ˜ µ considered in Example 4.2 is minimal. Proposition 4.14. Assume that µ ∈ M x ( G ) be definable, invariantly supported,idempotent and minimal (i.e. I µ = S ( µ ) ). Let ϕ ( x ) ∈ L x ( G ) be any formula. Thenfor any p, q ∈ S ( µ ) , we have that D ϕ ′ µ ( p ) = D ϕ ′ µ ( q ) . Proof. By Fact 4.11, S ( µ ) = S { i ∗ S ( µ ) : i ∈ J ( I µ ) } . By continuity, D ϕ ′ µ attains amaximum at some p ∈ S ( µ ). Let now q ∈ S ( µ ) = I µ be arbitrary. Then q ∈ i ∗ I µ for some i ∈ J ( I µ ). Also i ∗ p ∈ i ∗ I µ as I µ = S ( µ ). As i ∗ I µ is a group by Fact4.11(3), there exists some r ∈ i ∗ I µ such that r ∗ ( i ∗ p ) = q . But then, applyingProposition 4.6, we have D ϕ ′ µ ( p ) = D ϕ ′ µ (( r ∗ i ) ∗ p ) = D ϕ ′ µ ( r ∗ ( i ∗ p )) = D ϕ ′ µ ( q ) . As q ∈ S ( µ ) was arbitrary, this shows the proposition. (cid:3) Proposition 4.15. Assume that µ ∈ M x ( G ) be definable, invariantly supported,idempotent and minimal. Then for every ϕ ( x ) ∈ L x ( G ) , µ ( ϕ ( x )) = D ϕ ′ µ ( p ) for any p ∈ S ( µ ) .Proof. Assume not. By Proposition 4.14 and replacing ϕ ( x ) by ¬ ϕ ( x ) if necessary,we may assume that µ ( ϕ ( x )) > D ϕ ′ µ ( i ), where i is an idempotent in S ( µ ). Then µ ( ϕ ( x ) ∧ ¬ ϕ ( x · b )) > 0, where b | = i | G and G ≺ G is chosen as usual. Hence thereexists q ∈ S ( µ ) such that ϕ ( x ) ∧ ¬ ϕ ( x · b ) ∈ q . Then ϕ ( x ) ∈ q , and ¬ ϕ ( x ) ∈ q ∗ i . However, q ∗ i = q by Fact 4.11(2), and so we have ϕ ( x ) , ¬ ϕ ( x ) ∈ q — acontradiction. (cid:3) A direct translation of the previous proposition then says that minimal idempotentmeasures are “generically” right-invariant on their supports. Corollary 4.16. Assume that µ ∈ M x ( G ) be definable, invariantly supported,idempotent and minimal. Let ϕ ( x ; b ) ∈ L x ( G ) . Then, for any a ∈ G such that tp ( a/Gb ) ∈ S ( µ | Gb ) , we have µ ( ϕ ( x )) = µ ( ϕ ( x · a )) . Finally, we record a corollary for the case when the group G is stable and abelian. Remark 4.17. I µ = S ( µ ) if and only if for every p, q in the S ( µ ) there exists r ∈ S ( µ ) such that r ∗ q = p .The following corollary is a direct consequence of Glicksberg’s theorem for semi-topological semigroups [11] (note that unless the group is abelian, we only havecontinuity of ∗ on the left, so we were not in the context of Glicksberg’s theoremin the earlier considerations). Corollary 4.18. If G is stable, abelian and µ ∈ M x ( G ) is idempotent, then sup( µ ) is a compact Hausdorff topological group.Proof. Note that µ is automatically dfs by Fact 2.2(3), hence the results of thissection apply to it. We see that ( S ( µ ) , ∗ ) is commutative, as in Proposition 3.21.Then ∗ is both left and right-continuous. Hence I µ = S ( µ ) by Theorem 4.7. Butthis is equivalent to: for every p, q ∈ S ( µ ) there exists r ∈ S ( µ ) such that r ∗ q = p .By commutativity of ∗ and Fact 4.11, this implies that S ( µ ) is a group. Finally, bya classical theorem of Ellis [8], separate continuity of multiplication implies jointcontinuity for (locally) compact groups. (cid:3) Using this corollary, we can quickly describe idempotent measures in strongly min-imal groups. Example 4.19. Let G be a strongly minimal group. Then the idempotent measuresare precisely of the following form:(1) Haar measures on finite subgroups of G ; EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 19 (2) δ p , where p is the unique non-algebraic type in S x ( G ). Proof. Assume that G is ω -stable and abelian, and let µ be an idempotent measure.By Fact 2.2(3c) µ = P i ∈ ω r i · p i for some p i ∈ S x ( G ) and some r i ∈ R ≥ with P i ∈ ω r i = 1. By Corollary 4.18, S ( µ ) = { p i : i ∈ ω } is a countable compact group,and every countable compact group must be finite (using existence of finite Haarmeasure). So in fact µ = P i 1. Then,using Proposition 3.12, µ = µ ∗ µ = X ≤ i,j ≤ n n + 1 · δ a i · a j + 2 · nn + 1 · n + 1 · p + 1( n + 1) · p = X ≤ i,j ≤ n n + 1 · δ a i · a j + 2 n + 1( n + 1) · p. Consider the formula ϕ ( x ) := V ≤ i,j ≤ n x = a i · a j , then ϕ ( x ) ∈ p . Hence on theone hand µ ( ϕ ( x )) = n +1 , and on the other µ ( ϕ ( x )) = n +1( n +1) . But n +1 = n +1( n +1) = n +1( n +1) for any n ≥ 1, a contradiction. (cid:3) This example is generalized to arbitrary stable groups in the next section.5. Idempotent measures in stable groups In this section we classify idempotent measures on a stable group, demonstratingthat they are precisely the invariant measures on its type-definable subgroups. Ourproof relies on the results of the previous section and a variant of Hrushovski’sgroup chunk theorem due to Newelski [17]. We will assume some familiarity withthe theory of stable groups (see [21] or [27] for a general reference). As before, G isa monster model for a theory extending a group.5.1. Stabilizers of definable measures.Definition 5.1. Given a measure µ ∈ M x ( G ), we consider the following (left) stabilizer group associated to it:Stab( µ ) := { g ∈ G : g · µ = µ } = { g ∈ G : µ ( ϕ ( x )) = µ ( ϕ ( g · x )) for all ϕ ( x ) ∈ L ( G ) } . Below we use the characterization of definability of a measure from Fact 2.3(3),and we follow the notation there. Definition 5.2. Assume that µ x ∈ M x ( G ) is definable over a small model G ≺ G .(1) Fix a formula ϕ ( x ; y ) ∈ L and n ∈ N > . We write ϕ ′ ( x ; y, z ) to denote theformula ϕ ( z · x ; y ), and given i ∈ I n we writeΦ ϕ ′ , n ≥ i ( y, z ) := _ j ∈ I n ,j ≥ i Φ ϕ ′ , n j ( y, z ) . (2) Consider the following formula with parameters in G (where e is the identityof G ): Stab ϕ, n µ ( z ) := ∀ y ^ i ∈ I n ,i ≥ n (cid:16) Φ ϕ ′ , n ≥ i ( y, e ) → Φ ϕ ′ , n ≥ ( i − n ) ( y, z ) ∧ Φ ϕ ′ , n ≥ i ( y, z ) → Φ ϕ ′ , n ≥ ( i − n ) ( y, e ) (cid:17) . (3) We define the following partial type over G :Stab µ ( z ) := ^ ϕ ( x,y ) ∈L ,n ∈ N > Stab ϕ, n µ ( z ) . Proposition 5.3. Let µ ∈ M x ( G ) be definable. Then Stab( µ ) = Stab µ ( G ) .Proof. Assume g / ∈ Stab( µ ). Then there exist some ϕ ( x, y ) ∈ L and b ∈ G y suchthat taking r := µ ( ϕ ( x, b )) = µ ( ϕ ′ ( x ; b, e )) and s := µ ( ϕ ( g · x, b )) = µ ( ϕ ′ ( x, b, g )) wehave r = s . Say r > s (the case r < s is similar). We choose n ∈ N > large enoughso that | r − s | ≥ n (so in particular r ≥ n ). As { Φ ϕ ′ , n i ( G ) : i ∈ I n } is a coveringof G yz by Fact 2.3(3a), there is some i ∈ I n such that | = Φ ϕ ′ , n i ( b, e ), so particular | = Φ ϕ ′ , n ≥ i ( b, e ). Hence | r − i | < n by Fact 2.3(3b) (hence i ≥ n ). If | = Φ ϕ ′ , n ≥ ( i − n ) ( b, g ),then by Fact 2.3(3b) again we must have µ ( ϕ ′ ( x ; b, g )) > i − n − n , so s > i − n ,and r − s < n , contradicting the choice of n . Hence g = Stab ϕ, n µ ( z ).Assume g ∈ Stab( µ ), and let ϕ ( x, y ) , b ∈ G y , n ≥ i ≥ n in I n be arbitrary.Assume that | = Φ ϕ ′ , n ≥ i ( b, e ) holds, then by Fact 2.3(3b) we have µ ( ϕ ′ ( x ; b, e )) >i − n . If | = ¬ Φ ϕ ′ , n ≥ ( i − n ) ( b, g ), as { Φ ϕ ′ , n i ( G ) : i ∈ I n } is a covering, we must have | = Φ ϕ ′ , n j ( b, g ) for some j < i − n in I n . But then µ ( ϕ ′ ( x ; b, g )) < j + n byFact 2.3(3b) again. Hence µ ( ϕ ′ ( x ; b, g )) < i − n < µ ( ϕ ′ ( x ; b, e )), contradicting g ∈ Stab( µ ). Similarly, we get that | = Φ ϕ ′ , n ≥ i ( b, g ) implies | = Φ ϕ ′ , n ≥ ( i − n ) ( b, e ), hence g | = Stab ϕ, n µ ( z ) as wanted. (cid:3) Stable groups and group chunks. As before, T is a theory extending agroup in a language L , and we let G be a monster model of T . In this section wereview some results on stable groups that will be needed for our purpose. Fact 5.4. (see e.g. [20, Fact 1.8] + [3] ) Let G be a stable group and G ≺ G asmall model. Let H be a subgroup of G type-definable over G (by a partial type H ( x ) over G ). Let S H ( G ) := { p ∈ S x ( G ) : p ( x ) ⊢ H ( x ) } be the set of types over G concentrated on H . Then the following hold.(1) For p, q ∈ S H ( G ) , we have that p ∗ q is equal to tp( a · b/ G ) , where a | = p, b | = q and a | ⌣ G b (in the sense of forking independence).(2) The semigroup ( S H ( G ) , ∗ ) has a unique minimal closed left ideal I (also theunique minimal closed right ideal) which is already a subgroup of ( S H ( G ) , ∗ ) .(3) I is precisely the generic types of H , and with its induced topology I is acompact topological group (isomorphic to H / H ).(4) H admits a unique left invariant Keisler measure µ (which is also the uniqueright invariant Keisler measure) with S ( µ ) = I . Viewing µ as a regularBorel measure on S H ( G ) and restricting to the closed set I , µ ↾ S ( µ ) coincideswith the Haar measure on I . EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 21 In what follows, we work in the stable theory T G in the language L G with allof the elements of G named by new constants (obviously, T stable implies T G isstable), and let ˆ G ≻ G be a larger monster model of T G . We will be following thenotation from [17]. Definition 5.5. (1) Throughout this section, ∆ will denote a finite invariantset of formulas , i.e. formulas of the form ϕ ( u · x · v, ¯ y ) ∈ L G (so a right orleft translate of an instance of ϕ is again an instance of ϕ ).(2) We write R ∆ to denote Shelah’s ∆-rank, note that it is invariant undertwo-sided translation since ∆ is.(3) For P ⊆ S x ( G ), we let cl( P ) denote the topological closure of P , and ∗ P denote the closure of P under ∗ .(4) For P ⊆ S x ( G ), let gen( P ) denote the set of r ∈ cl( ∗ P ) such that there isno q ∈ cl( ∗ P ) with R ∆ ( r ) ≤ R ∆ ( q ) for all ∆ and R ∆ ( r ) < R ∆ ( q ) for some∆.(5) For P ⊆ S x ( G ), let h P i denote the smallest G -type definable subgroup of ˆ G containing P ( ˆ G ), where P ( ˆ G ) = { b ∈ ˆ G : b | = p for some p ∈ P } .In the following two facts, G is viewed as a small elementary submodel of thestable group ˆ G | = T G . Fact 5.6. (1) [17, Fact 2.1] If P ⊆ S x ( G ) is non-empty, then gen( P ) is a non-empty closed subset of S x ( G ) .(2) [17, Lemma 2.2] R ∆ ( p ∗ q ) ≥ R ∆ ( p ) , R ∆ ( q ) for any p, q ∈ S x ( G ) and ∆ (this follows by the symmetry of forking, invariance of R ∆ under two-sidedtranslations, and the fact that forking is characterized by drop in rank). The following fact is [17, Theorem 2.2] applied in T G . It is stated there for strongtypes over ∅ , which implies our statement as the elements of a small model G ≺ ˆ G are named by constants. Fact 5.7. ( T stable) Let P ⊆ S x ( G ) be non-empty set of types. Then h P i = n a ∈ ˆ G : tp( a/ G ) ∗ gen( P ) = gen( P ) setwise o is a G -type definable subgroup of ˆ G and gen( P ) is precisely the set of generic typesof h P i over G . Classification of idempotent measures. We are ready to prove the mainresult of this section. Theorem 5.8. Let G be a monster model of T , and let µ ∈ M x ( G ) be a globalKeisler measure (in particular, µ is dfs by Fact 2.2(3a)). Then the following areequivalent:(1) µ is idempotent;(2) µ is the unique right-invariant (and also the unique left-invariant) measureon the type-definable subgroup Stab( µ ) of G .Proof. (2) implies (1) by Proposition 3.17, and we show that (1) implies (2).Let µ ∈ M x ( G ) be an idempotent measure, by Fact 2.2(3a) µ is definable oversome small model G (cid:22) G by Proposition 5.3.By Corollary 4.4, S ( µ ) is a closed subset of S x ( G ) and is closed under ∗ , hencecl( ∗ S ( µ )) = S ( µ ) and gen( S ( µ )) ⊆ S ( µ ). We claim that gen( S ( µ )) is a two-sided ideal in ( S ( µ ) , ∗ ). Indeed, let r ∈ gen( S ( µ )) and q ∈ S ( µ ) be arbitrary. If r ∗ q is not in gen( S ( µ )), then thereexists some p ∈ S ( µ ) with R ∆ ( p ) ≥ R ∆ ( r ∗ q ) ≥ R ∆ ( r ) for all ∆ and some inequal-ity strict (by Fact 5.6(2)), contradicting r ∈ gen( S ( µ )). But also if q ∗ r is not ingen( S ( µ )), then there exists some p ∈ S ( µ ) with R ∆ ( p ) ≥ R ∆ ( q ∗ r ) ≥ R ∆ ( r ) andsome inequality strict, again by Fact 5.6(2), contradicting r ∈ gen( S ( µ )). Hencegen( S ( µ )) = S ( µ ) by Theorem 4.7.We now fix a larger monster model ˆ G ≻ G as above (and view G as a smallelementary submodel of it). Then, by Fact 5.7, we have thatˆ H := h S ( µ ) i = { a ∈ ˆ G : a | = p for some p ∈ S ( µ ) } is a G -type-definable subgroup of ˆ G and S ( µ ) = gen( S ( µ )) is precisely the set ofgeneric types of ˆ G restricted to G . Note that the definition of ˆ H a priori uses allof the parameters in G , and we need to argue that it can be defined over a subsetof G that is small with respect to G . Let H ( x ) be a partial type over G definingˆ H , i.e. H ( ˆ G ) = ˆ H . Given p ∈ S x ( G ), we let ˆ p ∈ S x ( ˆ G ) be its unique G -definableextension, and let ˆ µ ∈ M x ( ˆ G ) be the unique G -definable extension of µ (by Fact2.4). We have the following sequence of observations.(1) p ∗ q = r ⇐⇒ b p ∗ b q = b r for any p, q, r ∈ S x ( G ).(2) The same holds for measures, in particular b µ is an idempotent of (cid:16) M x ( b G ) , ∗ (cid:17) .Indeed, assume µ, ν ∈ M x ( G ) are definable over G . Then b µ ∗ b ν is definableover G (by Proposition 3.13) and extends µ ∗ ν , hence b µ ∗ b ν = [ µ ∗ ν byuniqueness of definable extensions (Fact 2.4).(3) Stab µ ( b G ) = Stab( b µ ) (by Proposition 5.3 and definability of the measure).(4) S ( b µ ) = { b p : p ∈ S ( µ ) } .Indeed, suppose p ∈ S ( µ ), but b p / ∈ S ( b µ ), then there is some ϕ ( x, b ) ∈ b p suchthat b µ ( ϕ ( x, b )) = 0. That is, | = d p ϕ ( b ), where d p ( y ) ∈ L y ( c ) for some finitetuple c ⊆ G is a ϕ -definition for p . By | G | + -saturation of G , we can findsome b ′ ∈ G y with b ′ ≡ Gc b . By definability (and hence invariance) of b µ over G , we have | = d p ( b ′ ) and b µ ( ϕ ( x, b ′ )) = µ ( ϕ ( x, b ′ )) = 0. So ϕ ( x, b ′ ) ∈ p ,contradicting p ∈ S ( µ ).Conversely, suppose r ∈ S ( b µ ). As b µ is definable over G , in particular it isnon-forking over G , hence every type in its support is non-forking over G .In particular r is definable over G , r = [ ( r | G ) and r | G is clearly in S ( µ ).(5) The generics of H ( x ) over b G are precisely { b p : p is a generic of H over G} .By stability, every generic r of H ( x ) over b G does not fork over G , so itis definable over G and r | G is a generic of H ( x ) over G , hence r = [ ( r | G ).Conversely, a definable (non-forking) extension of a generic type is generic.(6) Hence S ( b µ ) is precisely the set of the generics of H ( x ) over b G , in particular( S ( b µ ) , ∗ ) is a topological group by Fact 5.4(3).(7) Then b µ restricted to ( S ( b µ ) , ∗ ) (viewed as a regular Borel measure) is right ∗ -invariant.By (6), ( S ( b µ ) , ∗ ) is a group, so for any p ∈ S ( b µ ), p − is well-defined. Byregularity, it suffices to check ∗ -invariance for formulas. Let ϕ ( x, ¯ b ) ∈ L x ( b G ).Then for any p ∈ S ( b µ ) we have b µ ( ϕ ( x, ¯ b ) ∗ p ) = b µ (cid:0) { q ∗ p : ϕ ( x, ¯ b ) ∈ q } (cid:1) EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 23 = b µ (cid:0) { q : ϕ ( x, ¯ b ) ∈ q ∗ p − } (cid:1) = b µ ( ϕ ( x · c, ¯ b )) , where c | = p − | G ¯ b . And by Corollary 4.16, b µ ( ϕ ( x · c, ¯ b )) = b µ ( ϕ ( x, ¯ b )).(8) By Fact 5.4(4) for b H , there is a unique right b H -invariant Keisler measure ν ∈ M x ( b G ) such that ν ( H ( x )) = 1, S ( ν ) is the set of generics of H ( x ) over b G , and ν ↾ S ( ν ) (viewed as a Borel measure) is the Haar measure on thecompact topological group ( S ( ν ) , ∗ ).(9) Thus S ( b µ ) = S ( ν ), and as both µ, ν are right ∗ -invariant, by uniqueness ofthe Haar measure we have b µ ↾ S ( b µ ) = ν ↾ S ( ν ) , hence b µ = ν .(10) In particular, b H = Stab( ν ) = Stab( b µ ) = Stab µ ( b G ), so µ is the right invari-ant measure on the G -type definable group Stab µ ( x ). (cid:3) Remark 5.9. Some of these results can be generalized for idempotent measures inNIP groups, and we hope to address it in future work.6. Describing the convolution semigroup on finitely satisfiablemeasures as an Ellis Semigroup Dynamics. We begin this section by recalling the construction of the Ellissemigroup. Let X be a compact Hausdorff space and S be a semigroup acting on X by homeomorphisms. In particular, there is a map π : S × X → X such thatfor each s ∈ S , the map π s : X → X, x π ( s, x ) is a homeomorphism. Let X X be the space of functions from X to X equipped with the product topology. Then, { π s : s ∈ S } is naturally a subset of X X . Finally, the Ellis semigroup of the action ( X, S, π ) is (cl ( { π s : s ∈ S } ) , ◦ ), where we take the closure of { π s : s ∈ S } in X X .When the action map π is be clear, we will denote this semigroup as E ( X, S ).Let now T be a first order theory expanding a group, G a saturated model of T ,and G ≺ G a small elementary substructure. Recall that S x ( G , G ) denotes the setof global types finitely satisfiable in G . There is a natural action of G on S x ( G , G )via g · p = { ϕ ( x ) : ϕ ( g − · x ) ∈ p } . Fact 6.1 (Newelski [18]) . There exists a semigroup isomorphism (which is also ahomeomorphism of compact spaces) E ( S x ( G , G ) , G ) ∼ = ( S x ( G , G ) , ∗ ) . In this section, we provide an analogous description for the convolution semi-group on finitely satisfiable measures in NIP theories. Recall that M x ( G , G ) ⊆ M x ( G ) is the collection of global measures finitely satisfiable in G , and this spaceof measures carries a natural structure of a real topological vector space inducedfrom M x ( G ). We identify G with the set { δ g : g ∈ G } ⊆ M x ( G , G ), and let conv( G )denote the convex hull of G . There is a natural semigroup action of conv( G ) on M x ( G , G ): for any P ni =1 r i δ g i ∈ conv( G ) (with g i ∈ G and r i ∈ R ≥ , P ni =1 r i = 1), µ ∈ M x ( G , G ) and ϕ ( x ) ∈ L x ( G ), we define ( P ni =1 r i δ g i ) · µ ∈ M x ( G , G ) by n X i =1 r i δ g i ! · µ ! ( ϕ ( x )) = n X i =1 r i µ ( ϕ ( g i · x )) . For the rest of this section, we will denote elements of conv( G ) as k , the semi-group action described above as π : conv( G ) × M x ( G , G ) → M x ( G , G ), and the map µ π ( k, µ ) as π k . It is not difficult to see that for every k ∈ conv( G ), the map π k is continuous. Therefore, we can consider the Ellis semigroup of this semigroupaction, namely E ( M x ( G , G ) , conv( G )). We will show that if T is NIP, then this Ellis semigroup E ( G , G ) is isomorphicto the convolution semigroup of global measures which are finitely satisfiable in G ,i.e. ( M x ( G , G ) , ∗ ) (Theorem 6.10). We demonstrate that that these two semigroupsare isomorphic by considering the map ρ : M x ( G , G ) → M x ( G , G ) M x ( G ,G ) definedby ρ ( ν ) = ρ ν := ν ∗ − , and proving that the image of ρ is precisely the Ellissemigroup. Before continuing, we observe that ρ is well-defined, and that M x ( G , G )is a semigroup by recalling the following facts. Fact 6.2. Let T be NIP and assume that µ ∈ M x ( G , G ) . Then:(1) µ is Borel-definable over G (by Fact 2.2(2a));(2) for any ν ∈ M x ( G , G ) , µ ∗ ν ∈ M x ( G , G ) (by Proposition 3.13(2));(3) the operation ∗ on M x ( G , G ) is associative, hence ( M x ( G , G ) , ∗ ) is a semi-group (by associativity of ⊗ ). Hence the map ρ : M x ( G , G ) → M x ( G , G ) M x ( G ,G ) is well-defined. In the nextsubsection we show that it is also left-continuous.6.2. Left-continuity of convolution. We begin with a general continuity resultin arbitrary NIP theories. Let T be an NIP theory, U a monster model of T , and M a small elementary substructure of U . Proposition 6.3 (T NIP) . Let M ≺ U and let M inv x ( U , M ) be the closed set ofglobal M -invariant measures (Definition 2.1). If ν ∈ M y ( U ) and ϕ ( x ; y ) is anypartitioned L xy ( U ) formula, then the map − ⊗ ν ( ϕ ( x ; y )) : M inv x ( U , M ) → [0 , iscontinuous.Proof. Choose N ≺ U small and such that M (cid:22) N , and N contains the param-eters of ϕ . Then, choose a small N ≺ U such that N (cid:22) N and there exists someˆ ν ∈ M y ( U ) such that ˆ ν | N = ν | N and ˆ ν is smooth over N (by Fact 2.5). Fix ε ∈ R > , by Fact 2.2(1a) let b = ( b , . . . , b n ) be some ( ϕ ∗ , ε )-approximation for ˆ ν over N (where ϕ ∗ ( y ; x ) = ϕ ( x ; y ) and b is some element in ( N y ) <ω , see Definition2.1(7)). Note that every µ ∈ M inv x ( U , M ) is invariant over both N and N . Thenwe have (the last equality holds as in Proposition 3.3): µ ⊗ ν ( ϕ ( x ; y )) = Z S y ( N ) F ϕµ,N d ( ν | N ) = Z S y ( N ) F ϕµ,N d (ˆ ν | N ) = Z S y ( N ) F ϕµ,N d (ˆ ν | N ) . As ˆ ν is smooth over N , by Fact 3.7(1) we have Z S y ( N ) F ϕµ,N d (ˆ ν | N ) = Z S x ( N ) F ϕ ∗ ˆ ν,N d ( µ | N ) . Note that F ϕ ∗ Av b ,N ( p ) = n P ni =1 χ { r ∈ S x ( N ): ϕ ( x,b i ) ∈ r } ( p ) for every p ∈ S x ( N ),where χ is the characteristic function. Now, using that ¯ b ⊆ N is a ( ϕ ∗ , ε )-approximation for ˆ ν , we have the following (note that we identify ϕ ( x, b i ) withthe set of types satisfying it over N in the first step, and over U in the second step). Z S x ( N ) F ϕ ∗ ˆ ν,N d ( µ | N ) ≈ ε Z S x ( N ) F ϕ ∗ Av b ,N d ( µ | N )= Z S x ( N ) n n X i =1 χ ϕ ( x,b i ) ! d ( µ | N ) EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 25 = 1 n n X i =1 Z S x ( N ) χ ϕ ( x,b i ) d ( µ | N ) ! = 1 n n X i =1 µ | N ( ϕ ( x, b i ))= 1 n n X i =1 Z S x ( U ) χ ϕ ( x,b i ) dµ. Clearly, each map R χ ϕ ( x,b i ) : M x ( U ) → [0 , 1] is continuous by the definition of thetopology on the space of measures. Therefore, each map R χ ϕ ( x,b i ) : M inv x ( U , M ) → [0 , 1] is continuous, hence their sum is continuous as well. Since the choice of b isindependent of the choice of µ , we havesup µ ∈ M inv x ( U ,M ) | µ ⊗ ν ( ϕ ( x ; y )) − n n X i =1 Z S x ( U ) χ ϕ ( x,b i ) dµ | < ε. Therefore, the map − ⊗ ν ( ϕ ( x ; y )) is a uniform limit of continuous functions andhence continuous. (cid:3) Now, we apply this to our group theoretic context. Let again T be an NIP theoryexpanding a group, G a monster model of T , and G ≺ G a small model. Proposition 6.4. Let ν ∈ M x ( G , G ) . Then the map − ∗ ν : M x ( G , G ) → M x ( G , G ) is continuous.Proof. Let U be a basic open subset of M x ( G , G ). That is, there exist formulas ϕ ( x ) , ..., ϕ n ( x ) in L x ( G ) and real numbers r , ..., r n , s , ..., s n ∈ [0 , 1] such that U = n \ i =1 { µ ∈ M x ( G , G ) : r i < µ ( ϕ i ( x )) < s i } . Then we have (cid:16) − ∗ ν (cid:17) − ( U ) = n \ i =1 { µ ∈ M x ( G , G ) : r i < µ ∗ ν ( ϕ i ( x )) < s i } = n \ i =1 { µ ∈ M x ( G , G ) : r i < µ x ⊗ ν y ( ϕ i ( x · y )) < s i } = n \ i =1 (cid:0) − ⊗ ν y ( ϕ i ( x · y )) (cid:1) − (cid:16) { µ ∈ M x ( G , G ) : r i < µ ( ϕ i ( x )) < s i } (cid:17) . Therefore, by continuity of the map − ⊗ ν ( ϕ ( x · y )) (Proposition 6.3), the preimageof U under − ∗ ν is a finite intersection of open sets, and therefore open. (cid:3) The isomorphism. In this subsection we show that the map ρ : M x ( G , G ) →E ( G , G ) = E ( M x ( G , G ) , conv( G )) given by ρ ( ν ) = ρ ν = ν ∗ − is an isomorphism.We begin by recalling the topology on M x ( G , G ) M x ( G ,G ) . Remark 6.5. The topology on M x ( G , G ) M x ( G ,G ) is generated by the basic opensets of the form U = n \ i =1 { f : M x ( G , G ) → M x ( G , G ) | r i < f ( ν i )( ψ i ( x )) < s i } , with n ∈ N , r i , s i ∈ R , ψ i ( x ) ∈ L x ( G ), and ν i ∈ M x ( G , G ) (with possible repetitionsof ν i ’s and ψ i ’s). Lemma 6.6. The map ρ is injective.Proof. Note that for every ν ∈ M x ( G , G ), ρ ν ( δ e ) = ν , where e is the identity of G . (cid:3) Lemma 6.7. If µ ∈ M x ( G , G ) , then ρ µ ∈ cl (cid:0) { π k : k ∈ conv( G ) } (cid:1) . So ρ ( M x ( G , G )) ⊆ E ( M x ( G , G ) , conv( G )) .Proof. Let U be an open subset of M x ( G , G ) M x ( G ,G ) containing ρ µ . It is a union ofbasic open sets (see Remark 6.5), hence we can choose some n ∈ N , a sufficientlysmall ε > ψ ( x ) , ..., ψ n ( x ) ∈ L x ( U ) and ν , ..., ν n ∈ M x ( G , G ) such that B ε := n \ i =1 { f : | f ( ν i )( ψ i ( x )) − ρ µ ( ν i )( ψ i ( x )) | < ε } ⊆ U. Let H ≺ G be a small model containing G and the parameters of ψ , . . . , ψ n . ByFact 2.4, we can choose a small model H ≺ G and measures ˆ ν i ∈ M x ( G ) such that: • G (cid:22) H (cid:22) H ≺ G ; • ˆ ν i | H = ν i | H , for all 1 ≤ i ≤ n ; • ˆ ν i is smooth over H , for all 1 ≤ i ≤ n .Take some 0 < ε < ε . Recall from Section 3.2 that ψ ′ ( x ; y ) = ψ ( x · y ) ∈ L xy ( H ).By Fact 2.2(1a), let b i = ( b i,j : 1 ≤ j ≤ m i ) ∈ H <ω be a (( ψ ′ i ) ∗ , ε )-approximationfor ˆ ν i . Then, using that µ is invariant over both H and H and ˆ ν i is smooth over H as in Proposition 6.3, for every 1 ≤ i ≤ n we have: ρ µ ( ν i )( ψ i ( x )) = µ ∗ ν i ( ψ i ( x )) = µ ⊗ ν i ( ψ i ( x · y ))= Z S y ( H ) F ψ ′ i µ,H d ( ν i | H ) = Z S y ( H ) F ψ ′ i µ,H d (ˆ ν i | H )= Z S y ( H ) F ψ ′ i µ,H d (ˆ ν i | H ) = Z S x ( H ) F ( ψ ′ i ) ∗ ˆ ν i ,H d ( µ | H ) ≈ ε Z S x ( H ) F ( ψ ′ i ) ∗ Av bi ,H d ( µ | H ) = 1 m i m i X j =1 µ ( ψ i ( x · b i,j )) . Let Ψ = { ψ i ( x · b i,j ) : 1 ≤ i ≤ n, ≤ j ≤ m i } . Since µ is finitely satisfiable in G ,we can find some k µ ∈ conv( G ) such that k µ ( θ ( x )) = µ ( θ ( x )) for each θ ( x ) ∈ Ψ(see Proposition 2.11). We claim that then π k µ is in B ε . This follows directly fromrunning the equations above in reverse: for each 1 ≤ i ≤ n we have (using that k µ is obviously invariant over G , hence also over H )1 m i m i X j =1 µ ( ψ i ( x · b i,j )) = 1 m i m i X j =1 k µ ( ψ i ( x · b i,j ))= Z S x ( H ) F ( ψ ′ i ) ∗ Av bi ,H d ( k µ | H ) ≈ ε Z S x ( H ) F ( ψ ′ i ) ∗ ˆ ν i ,H d ( k µ | H )= Z S y ( H ) F ψ ′ i k µ ,H d (ˆ ν i | H ) = Z S y ( H ) F ψ ′ i k µ ,H d (ˆ ν i | H )= k µ ⊗ ν i ( ψ i ( x · y )) = π k µ ( ν i )( ψ i ( x )) . Hence ρ µ ( ν i )( ψ i ( x )) ≈ ε π k µ ( ν i )( ψ i ( x )) for each 1 ≤ i ≤ n , so π k µ ∈ B ε ⊆ U andwe are finished. (cid:3) Lemma 6.8. ρ ( M x ( G , G )) = E ( M x ( G , G ) , conv( G )) . EFINABLE CONVOLUTION AND IDEMPOTENT KEISLER MEASURES 27 Proof. Let f ∈ E ( M x ( G , G ) , conv( G )) be arbitrary. Then f ∈ cl ( { π k : k ∈ conv( G ) } ),and so there exists a net ( k i ) i ∈ I with k i ∈ conv( G ) such that lim i ∈ I π k i = f . Then,using Remark 6.5, for every ψ ( x ) ∈ L x ( G ) and ν ∈ M x ( G , G ) we havelim i ∈ I π k i ( ν )( ψ ( x )) = f ( ν )( ψ ( x )) . Consider δ e , where e ∈ G is the identity. Let µ f := f ( δ e ) ∈ M x ( G , G ). We claimthat the net ( k i ) i ∈ I converges to µ f in M x ( G , G ). Indeed, for any ψ ( x ) ∈ L x ( G ) wehave lim i ∈ I k i ( ψ ( x )) = lim i ∈ I π k i ( δ e )( ψ ( x )) = f ( δ e )( ψ ( x )) = µ f ( ψ ( x )) . Next, we claim that for any ν ∈ M x ( G , G ), we have that f ( ν ) = ρ µ f ( ν ). Indeed,first we have f ( ν ) = lim i ∈ I π k i ( ν ) = lim i ∈ I [ π k i ◦ ρ ν ]( δ e ) = lim i ∈ I ρ k i ∗ ν ( δ e ) = lim i ∈ I [ k i ∗ ν ] . The map − ∗ ν : M x ( G , G ) → M x ( G , G ) is continuous by Proposition 6.4, hence itcommutes with net limits. Therefore,lim i ∈ I [ k i ∗ ν ] = [lim i ∈ I k i ] ∗ ν = µ f ∗ ν = ρ µ f ( ν ) . We conclude that f = ρ µ f = µ f ∗ − . (cid:3) Lemma 6.9. The map ρ − : E ( M x ( G , G ) , conv( G )) → M x ( G , G ) is a continuousbijection.Proof. The map ρ − is a well-defined bijection by Lemmas 6.6 and 6.8. Let U bea basic open subset of M x ( G , G ), say U = n \ i =1 { µ ∈ M x ( G , G ) : r i < µ ( ϕ i ( x )) < s i } for some n ∈ N , ϕ i ( x ) ∈ L x ( U ) and r i , s i ∈ [0 , (cid:0) ρ − (cid:1) − ( U ) = n \ i =1 { f ∈ E ( M x ( G , G ) , conv( G )) : r i < f ( δ e )( ϕ i ( x )) < s i } . This is a restriction of a basic open subset (see Remark 6.5) to E ( M x ( G , G ) , conv( G )),hence open in the subspace topology. (cid:3) Theorem 6.10. The map ρ : ( M x ( G , G ) , ∗ ) → E ( M x ( G , G ) , conv( G )) is a homeo-morphism which respects the semigroup operation, and therefore an isomorphism.Proof. The map ρ is a homeomorphism since, by Lemma 6.9, ρ − is a continuousbijection between compact Hausdorff spaces. And note that ρ ( µ ∗ ν )( λ ) = ( µ ∗ ν ) ∗ λ = µ ∗ ( ν ∗ λ ) = ρ µ ( ν ∗ λ ) = ρ µ ◦ ρ ν ( λ ), hence ρ ( µ ∗ ν ) = ρ µ ◦ ρ ν . (cid:3) Remark 6.11. On the other hand, if T is NIP, then E ( M x ( G , G ) , G ) ∼ = E ( S x ( G , G ) , G ) , and so ∼ = ( S x ( G , G ) , ∗ ) by Fact 6.1. For a countable G ≺ G , this is an immediateconsequence of the corresponding observation in the context of tame metrizabledynamical systems (see e.g. [10, Theorem 1.5]); and for an arbitrary small G ≺ G , anapproximation argument with smooth measures (as in Lemma 6.7) can be adapted.As typically ( M x ( G , G ) , ∗ ) = ( S x ( G , G ) , ∗ ), we see that it was crucial to consider the action of conv( G ) rather than G in our characterization of ( M x ( G , G ) , ∗ ) as anEllis semigroup. References [1] Artem Chernikov. Model theory, Keisler measures, and groups. Bulletin of Symbolic Logic ,24(3):336–339, 2018.[2] Artem Chernikov, Anand Pillay, and Pierre Simon. 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