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APPEARED IN BULLETIN OF THE

arXiv:math/9304208v1 [math.LO] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 334-341BOREL ACTIONS OF POLISH GROUPSHoward Becker and Alexander S. KechrisAbstract. We show that a Borel action of a Polish group on a standard Borelspace is Borel isomorphic to a continuous action of the group on a Polish space,and we apply this result to three aspects of the theory of Borel actions of Polishgroups: universal actions, invariant probability measures, and the Topological VaughtConjecture.

We establish the existence of universal actions for any given Polish group,extending a result of Mackey and Varadarajan for the locally compact case. We provean analog of Tarski’s theorem on paradoxical decompositions by showing that theexistence of an invariant Borel probability measure is equivalent to the nonexistenceof paradoxical decompositions with countably many Borel pieces.We show thatvarious natural versions of the Topological Vaught Conjecture are equivalent witheach other and, in the case of the group of permutations of N, with the model-theoretic Vaught Conjecture for infinitary logic; this depends on our identification ofthe universal action for that group.A Polish space (group) is a separable, completely metrizable topological space(group).

A standard Borel space is a Polish space with the associated Borel struc-ture. A Borel action of a Polish group G on a standard Borel space X is an action(g, x) ∈G×X 7→g·x of G on X which is Borel, as a function from the space G×Xinto X.

The structure of Borel actions of Polish locally compact, i.e., second count-able locally compact, topological groups has long been studied in ergodic theory,operator algebras, and group representation theory. See, for example, [AM, Zi, Sin,VF, Mo, Ma1–Ma3, G, Var, FHM, Ra1, Ra2, K] for a sample of works related tothe themes that we will be studying here.

More recently, there has been increasinginterest in the study of Borel actions of nonlocally compact Polish groups. One in-stance is the Vaught Conjecture, a well-known open problem in mathematical logicand the Topological Vaught Conjecture (cf.

§§1, 2) (see, e.g., [Vau, Mi, St, Sa, L,Be, BM]). Another is the ergodic theory and unitary group representation theoryof so-called “large groups” (see, e.g., [Ve, O]).

Also [E] is relevant here.Our purpose in this note is to announce a number of results about Borel actionsof general Polish groups. With the exception of Theorem 2.1, these results are neweven for locally compact groups.

Our Theorem 2.1 is known in the locally compactcase [Ma2, Var], but the proofs in this case relied on Haar measure, so our proofseems new even in this case. The fundamental result is Theorem 1.1, stating thatfor the actions considered Borel actions are equivalent to continuous ones.1991 Mathematics Subject Classification.

Primary 03E15, 28D15.Received by the editors April 16, 1992 and, in revised form, October 15, 1992The first author’s research was partially supported by NSF Grant DMS-8914426. The secondauthor’s research was partially supported by NSF Grant DMS-9020153c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2HOWARD BECKER AND A. S. KECHRISThe paper is divided into three parts. The first part deals with Theorem 1.1 anda related result solving problems of Ramsay [Ra2] (raised in the locally compactcase) and Miller [Mi] and, as a direct application, shows the equivalence of threepossible versions of the Topological Vaught Conjecture.

In the second part, weestablish the existence of universal Borel actions for any Polish group, extendinga result of Mackey [Ma2] and Varadarajan [Var] from the locally compact case.This is also applied to establish the equivalence of the Topological Vaught Con-jecture for the symmetric group S∞, i.e., the permutation group of N, with theusual model-theoretic Vaught Conjecture for Lω1ω. The final section deals with theproblem of existence of invariant (countably additive) Borel probability measuresfor a Borel action of a Polish group.

It is a well-known theorem of Tarski (see[Wn]) that an arbitrary action of a group G on a set X admits a finitely additiveinvariant probability measure defined on all subsets of X iffthere is no “paradoxicaldecomposition” of X with finitely many pieces. We show that there is a completeanalog of Tarski’s theorem for Borel actions of Polish groups and countably additiveBorel invariant probability measures when we allow “paradoxical decompositions”to involve countably many Borel pieces.

Our proof uses the results of §1 and thebasic work of Nadkarni [N], who proves this result in case G = Z.1. Borel vs. topological group actionsBy a Borel G-space we mean a triple (X, G, α), where X is a standard Borelspace, G a Polish group, and α: G × X →X a Borel action of G on X.

We willusually write X instead of (X, G, α) and α(g, x) = g·x, when there is no dangerof confusion. Two Borel G-spaces X, Y are Borel isomorphic if there is a Borelbijection π: X →Y with π(g·x) = g·π(x).

A Polish G-space consists of a triple(X, G, α), where X is a Polish space and α: G × X →X is continuous (thus anyPolish G-space is also Borel).Theorem 1.1. Let X be any Borel G-space.

Then there is a Polish G-space YBorel isomorphic to X.This answers a question of [Ra2] (for locally compact G) and [Mi]. Theorem1.1 was known classically for discrete G and was proved in [Wh] for G = R. Aconvenient reformulation of our result is the following: If X is a Borel G-space,then there is a Polish topology τ on X giving the same Borel structure for whichthe action becomes continuous.

In our proof, sketched in §4, we define τ explicitlyand use a criterion of Choquet to show that it is a Polish topology.One canalso prove a version of Theorem 1.1 for more general “definable” actions of G onseparable metrizable spaces.It is a classical result of descriptive set theory (see, e.g., [Ku]) that for any Polishspace X and any Borel set B ⊆X, there is a Polish topology, finer than the topologyof X, and thus having the same Borel structure as X, in which B is clopen. Weextend this result in the case of Polish G-actions.

The result below was known forS∞(see [Sa]) and is essentially a classical result for discrete G.Theorem 1.2. Let X be a Polish G-space and B ⊆X an invariant Borel set.Then there is a Polish topology finer than the topology of X (and thus having thesame Borel structure) in which B is now clopen and the action is still continuous.The Topological Vaught Conjecture for a Polish group G, first conjectured by

BOREL ACTIONS OF POLISH GROUPS3Miller (see, e.g., [Ro, p. 484]) has been usually formulated in one of three, a prioridistinct, forms. Let:TVCI(G) ⇔For any Polish G-space X and any invariant Borel set B ⊆X,either B contains countably many orbits or else there is a perfect set P ⊆B withany two distinct members of P belonging to different orbits;TVCII(G) ⇔Same as TVCI(G) but with B = X;TVCIII(G) ⇔For any Borel G-space X either X contains countably many orbitsor else there is an uncountable Borel set P ⊆X with any two distinct members ofP belonging to different orbits.Clearly, TVCIII(G) ⇒TVCI(G), since a Borel set in a Polish space is uncount-able iffit contains a perfect set, and TVCI(G) ⇒TVCII(G).

From Theorem 1.1we have TVCII(G) ⇒TVCIII(G), thusCorollary 1.3. For any Polish group G, all three forms of the Topological VaughtConjecture for G are equivalent.The Topological Vaught Conjecture was motivated by the Vaught Conjecture inlogic, which we discuss in §2.

The truth or falsity of TVC(G) remains open forgeneral Polish G. It has been proved for G locally compact (see [Sil]) or abelian(see [Sa]). It can be also shown, using a method of Mackey, that if G is a closedsubgroup of H, then TVC(H) implies TVC(G).

Of particular interest is the caseG = S∞(see §2). It is known (see [Bu]) that for any Polish G-space X there areeither ≤ℵ1 many orbits or else perfectly many orbits, i.e., there is a perfect setwith any two distinct members of it belonging to different orbits.

So assuming thenegation of the Continuum Hypothesis (CH), there are 2ℵ0 many orbits iffthere areperfectly many orbits. Thus the meaning of TVC is that the set of orbits cannotbe a counterexample to CH.2.

Universal actionsLet X, Y be Borel G-spaces. A Borel embedding of X into Y is a Borel injectionπ: X →Y such that π(g·x) = g·π(x).

Note that π[X] is an invariant Borel subsetof Y . By the usual Schroeder-Bernstein argument, X, Y can be Borel-embeddedin each other iffthey are Borel isomorphic.A Borel G-space U is universal ifevery Borel G-space X can be Borel-embedded into U.

It is unique up to Borelisomorphism.Theorem 2.1. For any Polish group G, there is a universal Borel G-space UG.Moreover, UG can be taken to be a Polish G-space.Actually the proof of Theorem 2.1 shows that one can Borel-embed in UG anyBorel action of G on a separable metrizable space.

In particular, a Borel actionof G on an analytic Borel space (i.e., an analytic set with its associated Borelstructure) is Borel isomorphic to a continuous action of G on an analytic space,i.e., an analytic set with its associated topology. For locally compact G, Theorem2.1 has been proved in [Ma2, Var].

In this case, UG can be taken to be compact.Our proof gives a new proof of this result, with a different universal space, whichavoids using the Haar measure. It is unknown whether UG can be taken to becompact in the general case.In the particular case of the group G = S∞, with the Polish topology it inheritsas a Gδ subspace of the Baire space NN, our proof of Theorem 2.1 gives a particularly

4HOWARD BECKER AND A. S. KECHRISsimple form of UG, which we use below. Let X∞be the space of all maps from theset N

Clearly this space ishomeomorphic to the Cantor space. Consider the following action of S∞on X∞:g·x(s0, s1, .

. .

, sn−1) = x(g−1(s0), . .

. , g−1(sn−1)), if s = (s0, s1, .

. .

, sn−1) ∈N

The VaughtConjecture for Lω1ω is the assertion that every Lω1ω sentence in a countable lan-guage has either countably many or else perfectly many countable models, upto isomorphism.Let L be a countable language which we assume to be rela-tional, say L = {Ri}i∈I, where I is a countable set and Ri is a ni-ary relationsymbol.Let XL = Qi∈I 2Nni which is homeomorphic to the Cantor space (ifL ̸= ∅). We view XL as the space of countably infinite structures for L, identifyingx = (xi)i∈I ∈XL with the structure A = ⟨N, RAi ⟩, where RAi (s) ⇔xi(s) = 1.

Thegroup S∞acts in the obvious way on XL : g·x = y ⇔∀i[yi(s0, . .

. , sni−1) = 1 iffxi(g−1(s0), .

. .

, g−1(sni−1)) = 1]. Thus if x, y are identified with the structures A,B resp., g·x = y iffg is an isomorphism of A, B.

This action is called the logicaction of S∞on XL. By [L-E] the Borel invariant subsets of this action are exactlythe sets of models of Lω1ω sentences.

So the Vaught Conjecture for Lω1ω (VC), anotorious open problem, is the assertion that for any countable L and any Borelinvariant subset B ⊆XL, either B contains countably many orbits or else there isa perfect set P ⊆B, no two distinct members of which are in the same orbit. Itis thus a special case of the Topological Vaught Conjecture for S∞.

(Historically,of course, TVC came much later and was inspired by VC.) The universal BorelS∞-space X∞above is the same as XL∞for L∞= {Rn}n∈N, with Rn an n-aryrelation, so we haveCorollary 2.2.

The logic action on the space of structures of the language contain-ing an n-ary relation symbol for each n ∈N is a universal S∞-space. In particular,Vaught’s conjecture for Lω1ω is equivalent to the Topological Vaught Conjecture forS∞.We conclude with an application to equivalence relations.

Given two equivalencerelations E, F on standard Borel spaces X, Y resp., we say that E is Borel embed-dable in F iffthere is a Borel injection f : X →Y with xEy ⇔f(x)Ff(y). Given aclass of equivalence relations S, a member F ∈S is called universal if every E ∈Sis Borel embeddable in F. For each Borel G-space denote by EG the correspondingequivalence relation induced by the orbits of the action: xEGy ⇔∃g ∈G(g·x = y).The following is based on the proof of Theorem 2.1 and the work in [U], a paperwhich was brought to our attention by W. Comfort.Corollary 2.3.

There is a universal equivalence relation in the class of equivalencerelations EG induced by Borel actions of Polish groups.3. Invariant measuresLet G be a group acting on a set X.Given A, B ⊆X, we say that A, Bare equivalent by finite decomposition, in symbols A ∼B, if there are partitionsA = Sni=1 Ai, B = Sni=1 Bi, and g1, .

. .

, gn with gi.Ai = Bi. We say that X isG-paradoxical if X ∼A ∼B with A ∩B = ∅.

A finitely additive probability (f.a.p.)

BOREL ACTIONS OF POLISH GROUPS5measure on X is a map ϕ: P(X) →[0, 1] such that ϕ(X) = 1, ϕ(A) + ϕ(B) =ϕ(A ∪B), if A ∩B = ∅. Such a ϕ is G-invariant if ϕ(A) = ϕ(g·A) for all g ∈G andA ⊆X.

If X is G-paradoxical, there can be no G-invariant f.a.p. measure on X.A well-known theorem of Tarski asserts that the converse is also true (see [Wn]).Theorem 3.1 (Tarski).

Let a group G act on a set X. Then there is a G-invariantfinitely additive probability measure on X iffX is not G-paradoxical.It is natural to consider to what extent Tarski’s theorem goes through for count-ably additive probability measures.

Let (X, A) be a measurable space, i.e., a setequipped with a σ-algebra. Let a group G act on X so that A ∈A ⇒g·A ∈A.Given A, B ∈A, we say that A, B are equivalent by countable decomposition, insymbols A ∼∞B, if there are partitions A = Si∈I Ai, B = Si∈I Bi, with I count-able and Ai, Bi ∈A, and {gi}i∈I so that gi·Ai = Bi.

We say that (X, A), or justX if there is no danger of confusion, is countably G-paradoxical if X ∼∞A ∼∞Bwith A, B ∈A and A ∩B = ∅. A probability measure µ on (X, A) is G-invariant ifµ(A) = µ(g·A) for all g ∈G and A ∈A.

Again, if there is a G-invariant probabilitymeasure on (X, A), X cannot be countably G-paradoxical. Is the converse true?

Itturns out that the answer is negative in this generality. See [Wn, Za] for more onthe history and some recent developments on this problem.

We show here, however,that the converse holds in most regular situations, i.e., when G is Polish and actsin a Borel way on a standard Borel space X.Theorem 3.2. Let X be a Borel G-space.

Then the following are equivalent:(1) X is not countably G-paradoxical;(2) there is a G-invariant Borel probability measure on X.The proof of this theorem is based on the results in §1 and [N], which provesTheorem 3.2 for the case G = Z. It turns out that Theorem 3.2 holds also for anycontinuous action of a separable topological group on a Polish space X.4.

Sketches of proofsFor Theorem 1.1. We have a Borel action of G on X.

Fix a countable basis B forthe topology of G. For A ⊆X and U ⊆G open, let A∆U = {x : g·x ∈A for aset of g’s which is nonmeager in U} be the Vaught transform [Vau]. We first find acountable Boolean algebra C of Borel subsets of X such that (1) A ∈C ⇒A∆U ∈Cfor U ∈B, and (2) the topology generated by C is Polish.

Then let τ be the topologyon X generated by {A∆U : A ∈C, U ∈B}. It suffices to show that τ is a Polishtopology and that the action of G is continuous in this topology.

Then since τconsists of Borel sets in X and gives a Polish topology, it gives rise to the originalBorel structure of X.The continuity of the action may be checked by direct computation.To seethat τ gives a Polish topology, we first check that it is T1 and regular, as well asobviously second countable, hence metrizable, and then apply a criterion of Choquetto conclude that it is Polish. Choquet’s criterion may be stated as follows.Associate to X the strong Choquet game, in which the first player specifies asequence of open sets Un and elements xn ∈Un, while the second player respondswith open sets Vn, satisfying xn ∈Vn ⊆Un and Un+1 ⊆Vn; the second player winsif the intersection of the Un is nonempty.

The space X is called a strong Choquetspace (see, e.g., [HKL]) if the second player has a winning strategy for this game,

6HOWARD BECKER AND A. S. KECHRISand Choquet’s criterion (unpublished, but see [C] for a related version) states thata topological space is Polish if and only if it is separable, metrizable, and strongChoquet.For Theorem 1.2. The proof is quite similar, taking a somewhat larger C and in-cluding a basis for the original topology in τ.For Theorem 2.1.

Let F(G) be the space of closed subsets of G with the (standard)Effros Borel structure, i.e., the one generated by the sets FV = {F ∈F(G): F ∩V ̸=∅}, for V ⊆G open. Let G act on F(G) by left multiplication and let UG = F(G)N,with G acting coordinatewise.

Given any Borel G-space X, let {Sn} be a sequenceof Borel subsets of X separating points. For each A ⊆G, let D(A) = {g ∈G: Forevery neighborhood U of g, A is nonmeager in U} and map x ∈X into the sequencef(x) = {D(eSn)−1} ∈UG, where eSn = {g : g·x ∈Sn}.

This is an embedding of Xinto UG. We can now make UG into a Polish G-space using Theorem 1.1.

When Gis locally compact and G = G∪{∞} is the one-point compactification of G, then wecan extend the action of G on itself by left multiplication to G by setting g·∞= ∞.Then instead of F(G) we can use F(G) = K(G) =the compact metrizable space ofcompact subsets of G, with the obvious G-action, and the universal space is nowK(G)N, which is compact. (Alternatively, we can use the Fell topology on F(G).

)For Corollary 2.2. The action of S∞on itself by left multiplication extends to theaction of S∞on NN (the Baire space) by left composition.

So instead of F(S∞),one can use F(NN) and thus can take F(NN)N as a universal space. But closedsubsets of NN can be identified with trees on N, i.e., subsets of N

A tree can be viewed now as a sequence of n-ary relations on N,the nth relation identifying which n-tuples belong to the tree. Thus F(NN)N canbe embedded in the space of structures XL, for the language containing infinitelymany n-ary relation symbols for each n, and then by a simple coding pointed outby R. Dougherty, in the space of structures XL∞for the language L∞.For Theorem 3.2.

One first shows that the result holds for countable groups byadapting appropriately the proof of [N]. For an arbitrary G, let G1 be a countabledense subgroup of G and notice that if G acts continuously, then any G1-invariantmeasure is also G-invariant.

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