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The automorphism group of the Gaussian measure cannot act pointwise

Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for Boolean G-actions. We show that in full generality this theorem does not hold for actions of Polish groups. In particular there is no Borel model for the Polish automorphism group of a Gaussian measure. In fact, we show that this group as well as many other Polish groups do not admit any nontrivial Borel measure preserving actions.


Introduction
Our motivation is threefold: invariant measures; Borel liftings; Gaussian measures.
Invariant measures: By a famous theorem of A. Weil if a Polish group G admits a σ-finite invariant measure then G is locally compact (see Appendix B). Nonetheless, even if G is not locally compact, a homogeneous space of G might even admit a finite invariant measure. For example, the group G of µpreserving homeomorphisms of the Cantor set Ω, will act transitively on Ω for a suitable choice of µ (see for example [8]). We show that this never happens for some classes of Polish groups G (for instance, the full unitary group of a separable Hilbert space), except for the trivial case: a measure concentrated on fixed points. Borel liftings: Let G be a closed subgroup of the Polish group of all invertible measure preserving transformations of (say) [0, 1] with Lebesgue measure. An element g ∈ G is an equivalence class of maps [0, 1] → [0, 1] rather than a single map; thus, g(x) is defined only almost everywhere. Can we define g(x) everywhere? More exactly: can we lift the mod 0 action to a Borel action? We give a general criterion for lifting, and a negative answer for some classes of groups including the Gaussian case. Gaussian measures: Every Euclidean space carries its standard Gaussian measure. However, a separable Hilbert space H (over R) does not. The standard Gaussian process over H is a linear isometry between H and the subspace spanned by a sequence of i.i.d. N(0, 1) random variables in L 2 over a probability space. Can we implement each point of (some version of) the probability space as a function on H or another superstructure over H? Some well-known such constructions are 'isotropic', that is, invariant under the full orthogonal group of H. Others give a standard probability space. We show that these two desirable properties exclude each other. Two proofs are given, one via 'invariant measures', the other via 'Borel liftings'. Having thus stated our goals in outline let us be more precise. Traditionally, ergodic theory is treated within the context of locally compact groups acting on standard Lebesgue probability spaces. However it is often the case that one has to deal with near-actions (see definition below) or merely with an action of the group on a measure algebra (i.e. the Borel algebra modulo sets of measure zero) and it is then desirable to find a standard Lebesgue model, or even better, a Polish (= complete metric, second countable space) or a compact model where the group acts continuously.
Recall that a Borel action of G on a Borel space (X, X) is a Borel map G × X → X (denote it just (g, x) → gx) satisfying the two conditions, ex = x and g(hx) = (gh)x for all g, h ∈ G and all x ∈ X. Such an object is called also a Borel G-space.
Definition 0.1. (See Zimmer [34,Def. 3.1]) Let G be a Polish group and (X, X, µ) a standard Borel space with a probability measure µ. By a near-action of G on (X, X, µ) we mean a Borel map G × X → X, (g, x) → gx with the following properties: (i) With e the identity element of G, ex = x for almost every x.
(ii) For each pair g, h ∈ G, g(hx) = (gh)x for almost every x (where the set of points x ∈ X of measure one where this equality holds may depend on the pair g, h). (iii) Each g ∈ G preserves the measure µ.
Let Aut (X) = Aut (X, X, µ) be the Polish group of all equivalence classes of invertible measure preserving transformations X → X, with the neighborhood basis at the identity formed by sets of the form N(A, ε) = {T ∈ Aut (X) : µ(A△T A) < ε}, for A ∈ X and ε > 0. What we would like to show next is that the following three notions are equivalent.
(I) A near-action of G on (X, X, µ). (II) A continuous homomorphism from G to Aut (X). (III) A Boolean action of G on (X, X, µ), that is, a continuous homomorphism from G to the automorphism group of the associated measure algebra. Given a near action of G, it is easy to check that the natural mapping from G to Aut (X) defines a measurable mapping. That it is a homomorphism follows from the defining property of being a near action, and since, as is well known, measurable homomorphisms of Polish groups are continuous, we get (II) from (I). To go in the other direction, we must construct from a continuous homomorphism of G into Aut (X), a near action of G on (X, X, µ).
For this we need to define a section on equivalence classes of Borel measurable functions where the equivalence relation is that of equality µ a.e. Let (X, X) and (Y, Y) be standard Borel spaces and µ a probability measure on (X, X). Then the set L 0 (X, Y ) = L 0 ((X, X, µ), (Y, Y)) of all equivalence classes (mod 0 with respect to µ) of Borel (or just µ-measurable) maps X → Y is also a standard Borel space; its σ-algebra is generated by functions There exists a (highly non-unique) Borel map V : L 0 (X, Y ) × X → Y such that for every f ∈ L 0 (X, Y ) the function x → V (f, x) belongs to the equivalence class f . For example, assuming X = Y = (0, 1) (with the usual Borel σ-algebra and Lebesgue measure µ), we may take The Polish group Aut (X) is a Borel subspace of L 0 (X, X). Returning now to our situation, let φ denote a continuous homomorphism of G into Aut (X). Composing φ with the restriction of V above to Aut (X) × X gives us a Borel mapping from G × X to X, and one checks easily that the properties for being a near action are satisfied. Thus (II) implies (I).
Finally, Aut (X) may be thought of as the automorphism group of the measure algebra MALG (X, X, µ) = (X,μ), whereX is X modulo nullsets, andμ the corresponding measure; automorphisms of the measure algebra must preserve Boolean operations andμ. Thus (II) and (III) are equivalent.
This completes the discussion of the equivalence of the various notions of a nearaction.
In contrast to near-actions, we define the notion of spatial action.
Definition 0.2. Let G be a Polish group. By a spatial G-action we mean a Borel action of G on a standard Lebesgue space (X, X, µ) such that each g ∈ G preserves the measure µ. We say that two spatial actions are isomorphic, if there exists a measure preserving one to one map between two G-invariant subsets of full measure in the corresponding spaces which intertwines the G-actions (the same two sets for all g ∈ G).
Every spatial action is also a near-action. In that case the spatial action will be called a spatial model of the near-action (or the corresponding Boolean action). The question is, when does a given near-action admit a spatial model. Rohlin,in [29], when discussing R-flows, distinguishes between these two notions calling our near-actions continuous flows and the spatial actions measurable flows. He notes there that the theories of these two notions are not equivalent. Indeed already J. von Neumann in his foundational work [24] was aware of these distinctions and in footnote 13 writes that he hopes to provide a proof that every near-flow has a continuous spatial model.
We recall that a Polish G-space is a Polish space X together with a continuous action G × X → X of a Polish group G. Such an action will be called a Polish action. If in addition X is compact then it is a compact Polish G-space.
Every Polish action is also a Borel action. In that case the Polish action will be called a Polish model of the Borel action.
In the present work we show that a full generalization to Polish groups of the first part of Theorem 0.3 is not possible. Many near-actions (or Boolean actions) of Polish groups admit no spatial models.
In Section 1 we recall the definition of Lévy groups, a class of groups which includes Aut (X, X, µ) and U(H), the unitary group on an infinite dimensional Hilbert space. We show that these groups admit no non-trivial spatial actions, and discuss some further examples. In Section 2 we take up the general question of finding criteria for a near-action to admit a spatial model, or more generally a spatial factor. We give a necessary and sufficient condition for this in terms of G-continuous functions. These are those functions f ∈ L ∞ (µ) with the property that f • g n converges to f in L ∞ (µ) norm whenever g n → e, the unit element in G.
In order to apply the criterion of Section 2 we introduce in Section 3 whirly actions which may be defined as near-actions of G on (X, X, µ) such that for all sets A, B ∈ X of positive measure, every neighborhood of e contains g with µ(A ∩ gB) > 0. We show that whirly actions have no non-constant G-continuous functions and verify some easy examples of whirly actions.
In Section 4 we return to the natural near-action of the Polish orthogonal group G = O(H), when H is the "first chaos" Gaussian Hilbert space and show that this near-action is whirly, thus giving the promised second proof of the non-existence of a spatial model for this action.
Appendix A collects elementary proofs of the Lévy property for many well known Lévy families. In Appendix B we provide a proof of the fact, mentioned at the beginning of the introduction, that a Polish group which admits a Borel σ-finite invariant measure is locally compact.
We thank A. Vershik, V. Milman and A. Kechris for instructive remarks concerning the historical background of this work. We thank V. Pestov for a careful reading of a draft of this paper and for several useful remarks.

Lévy groups admit no spatial actions
The phenomenon of concentration of mass was first considered by E. Borel in his "law of large numbers", where it is manifested in the family of "discrete cubes" {0, 1} n equipped with Hamming distance and counting measure. In the 30's P. Lévy studied the concentration phenomenon for the family of Euclidean spheres. Then about 1970 V. Milman in [20] revitalized the area when he discovered a new proof of Dvoretzky's theorem using the concentration phenomenon on spheres. In the work [11] M. Gromov and V. Milman considered applications of the concentration phenomenon in topological dynamics. In particular the notion of a Lévy family is introduced in [11]. See [23] and [27] for further details on the history of this subject.
Let (X n , d n , µ n ), n = 1, 2, 3 . . . be a family of metric spaces with probability measures µ n . Call such a family a Lévy family if the following condition is satisfied. If A n ⊂ X n is a sequence of subsets such that lim inf µ n (A n ) > 0 then for any ε > 0, lim µ(B ε (A n )) = 1, where B ε (A) is the ε neighborhood of A.
A Polish group G is a Lévy group if there exits a family of compact subgroups K n ⊂ K n+1 such that the group F = ∪ n∈N K n is dense in G and the corresponding family (K n , d, m n ) is a Lévy family; here m n is the normalized Haar measure on K n , and d is a right-invariant compatible metric on G (the choice of d does not matter).
Using left-invariant metrics instead, we get an equivalent definition (just apply the map g → g −1 ).
Theorem 1.1. Every spatial action of a Lévy group is trivial; i.e. the set of fixed points is of full measure.
Proof. By Theorem 0.4(b), every Borel G-space is embedded into a compact Polish G-space. Therefore it suffices to prove the theorem for a continuous action of G on a metrizable compact space X and a G-invariant Borel probability measure µ. We will see that G acts trivially on supp µ (the support of the measure). The action is a continuous homomorphism from G to the Polish group Homeo (X) of all homeomorphisms of X (as noted in [27, p. 427]). We equip X with a compatible metric ρ, and Homeo (X) with the compatible right-invariant metric (f, g) → max x∈X ρ(f (x), g(x)). Now the homomorphism is uniformly continuous, provided that G is also equipped with a right-invariant metric (which will be assumed).
The family (g → g · x) x∈X of maps G → X is equicontinuous. By [11, 2.1], it sends the Lévy family (m n ) of measures on G to a Lévy family (m n · x) of measures on X, uniformly in x ∈ X. In other words: for all x 1 , x 2 , · · · ∈ X the family (m n · x n ) is Lévy.
By [11, 2.4] the family of measures is degenerate in the sense that for n → ∞, uniformly in x ∈ X. The proof is simple. Assuming the contrary and using compactness, we choose x k ∈ X and n k → ∞ such that measures m n k · x k converge (weakly) to some measure ν on X satisfying min y∈X X ρ(·, y) dν > 0, which means that the support of ν contains at least two points. Every open set A ⊂ X such that ν(A) > 0 satisfies ν(B ε (A)) = 1 for all ε > 0; here B ε (A) is the closed ε-neighborhood of A. We get a contradiction by choosing A such that some points of the support of ν belong to A and some do not belong to the closure of A.
For each n we introduce the subspace H n ⊂ L 2 (µ) of all K n -invariant functions, and the corresponding orthogonal projection Q n , For f ∈ C(X) ⊂ L 2 (X) the degeneracy of measures gives We see this as follows. Denoting by y n,x the minimizer of X ρ(·, y) d(m n · x) we have Taking lim sup ε→0 lim sup n→∞ sup x∈X (. . . ) we get 0.
On the other hand, the integral does not depend on g ∈ K n and is equal to f − Q n f 2 . Therefore for all x ∈ supp µ and all f ∈ C(X). Thus, g · x = x for all such x and all g ∈ ∪K n , therefore all g ∈ G.
Question 1.2. Can a Lévy group admit a nontrivial nonsingular (that is, preserving a measure class) Borel action? Remark 1.3. The basic idea in the proof of Theorem 1.1 is derived from Gromov and Milman [11] where they show that Lévy groups have the fixed point on compacta property. The question arises whether every group with the fixed point property do not admit a nontrivial spatial measure preserving Borel action. Now it was shown by Pestov [26] that the Polish group G = Aut (Q, <) of order preserving permutations of the rational numbers, equipped with the topology of pointwise convergence (with respect to the discrete topology on Q), has the fixed point on compacta property (or is extremely amenable). However it is easy to see that this group also acts ergodically by homeomorphisms on the "Q-Bernoulli system" (Ω, F, µ).
Here Ω = {1, −1} Q , µ is the product measure µ = (1/2, 1/2) Q and G acts on "configurations" ω ∈ Ω by permuting the indices. We therefore conclude that some Polish groups with the fixed point property can have nontrivial spatial actions.
Remark 1.4. Recall that a topological group G is amenable if each compact G-space admits a G-invariant probability measure. Using this definition of amenability and the fact that every compact group is amenable it is easy to deduce that every Lévy group is amenable. Now if G is a Polish Lévy group and (X, G) is a compact Gspace, then by amenability of G there is a G-invariant probability measure µ on X.
It can be shown that (X, G) is represented as an inverse limit of a directed system of metrizable G-spaces {(X α , G)}. Let µ α be the image of µ on X α , then apply theorem 1.1 to deduce that supp (µ α ) is a closed nonempty collection of fixed points. It is now easy to conclude that the support of µ, supp (µ) is a nonempty closed subset of X consisting of fixed point. Thus the Gromov-Milman theorem that every Lévy group has the fixed point on compacta property follows from Theorem 1.1. Of course this is a rather circumventive way of proving it. and [2]. In the latter Banaszcyk provides, for every infinite dimensional normed space E, examples of the form G = E/K where K ⊂ E is a discrete subgroup. It is easy to see that any such group is moreover monothetic. Of course such "strongly exotic groups" as they are called by Herer, Christensen and Banaszcyk can not admit even a nontrivial near-action. Moreover, every nonsingular near-action (preserving a measure class rather than a measure) leads, by a standard construction, to a unitary representation. Thus these strongly exotic groups can not admit nontrivial nonsingular near-actions.
By Theorem 1.1, a nontrivial near-action of a Lévy group cannot admit a spatial model. An important example is the automorphism group of an infinite-dimensional Gaussian measure. Up to isomorphism, the relevant probability space is the product (R ∞ , γ ∞ ) of countably many copies of (R, γ), where γ is the standard one-dimensional Gaussian measure (normal distribution). The space (so-called first chaos) of all measurable linear functionals on (R ∞ , γ ∞ ) is l 2 . The action of the full orthogonal group O(l 2 ) on measurable linear functionals is well-known to be induced by its near-action on (R ∞ , γ ∞ ), which is what we mean by the automorphism group of the Gaussian measure. (In this sense, O(l 2 ) is a closed subgroup of Aut (R ∞ , γ ∞ ); see also Section 4.)  . The latter group contains (an isomorphic copy of) the former group as a closed subgroup, see e.g. Lemańczyk, Parreau and Thouvenot [17]. The near-action of (the copy of) L 0 [0, 1], S 1 on (R ∞ , γ ∞ ) is nontrivial; by Theorem 1.1 it cannot admit a spatial model, which implies Corollary 1.8.
Remark 1.11. About the meaning of Corollary 1.8. Almost all points of (R ∞ , γ ∞ ) do not belong to l 2 and therefore cannot be interpreted as continuous linear functionals on l 2 . One could hope for interpreting them as another superstructures over l 2 (say, densely defined discontinuous linear functionals) that form a Borel G-space (G being the symmetry group

Which actions admit spatial models ?
In this section we enhance our understanding of the lifting problem by relating it to a notion of G-continuity of functions which is reminiscent of the classical notion of a rigid action in ergodic theory.
Definition 2.1. Having a near-action (or Boolean action) of G on (X, µ) we say that The collection A(G) of all G-continuous functions is a G-invariant closed subalgebra of L ∞ (µ). Proof. Suppose first that we have a spatial model, that is, a Borel G-space with an invariant measure. By Theorem 0.4(b) this Borel G-space can be embedded into a compact Polish G-space X (with an invariant measure). The continuous functions on X form a separable Banach space and a dense sequence in C(X) will provide a sequence of G-continuous functions in L ∞ (µ) which separates points.
Conversely, suppose there exists a sequence {f n : n ∈ N} ⊂ L ∞ (µ) of G-continuous functions that generates the σ-algebra. Let G 0 ⊂ G be a countable dense subgroup of G. Let A ⊂ L ∞ (µ) be the smallest closed G 0 -invariant subalgebra containing {f n : n ∈ N} and the constant functions. Clearly A is a separable subalgebra and the fact that G 0 is dense in G implies that A is in fact G-invariant.
Let Y be the compact metric Gelfand space of A. (Thus the elements of Y are the multiplicative linear functionals of norm one on A and the map A ∼ = C(Y ), f →f , wheref (y) = y(f ), is an isometric isomorphism of Banach algebras.) Then, for each g ∈ G, the linear action f → f • g of g on A defines a homeomorphism g : Y → Y and f • g =f • g. If y n → y in Y and g n → e in G are convergent sequences then for everyf ∈ C(Y ) we have hence lim n→∞ |f (g n y n ) −f (y)| = 0. It follows that lim n→∞ g n y n = y and we conclude that the action of G on Y is topological.
The linear functional µ : A → R, f → f dµ defines a probability measure ν on Y and the dynamical system (Y, Y, ν, G), where Y is the Borel σ-algebra on Y , yields a Boolean action (Y, ν, G) which is isomorphic to the given Boolean action. We conclude that (Y, Y, ν, G) is a spatial model as required.
Remark 2.3. In general when we do not assume that the G-continuous functions on the near action (X, X, µ, G) separate points, we can still consider the smallest σ-algebra D ⊂ X with respect to which all the functions in A(G) are measurable and then the closed subspace of L 2 (µ) consisting of D-measurable functions. This subspace defines a factor near action and it is clear that this factor is the largest factor which admits a spatial model. Theorem 1.1 together with remark 2.3 yield the following:

Corollary 2.4. For a Lévy group G, an ergodic near-action admits only constants as G-continuous functions.
It is an interesting fact that a seemingly weaker condition already implies Gcontinuity. To see this we first need a lemma.
Lemma 2.5. Let X be a Polish space, f : X → L 1 (µ) a continuous map such that the image f (X) is contained in L ∞ (µ) and that as a subset of the Banach space L ∞ (µ) it is separable. Then f treated as a map X → L ∞ (µ) is continuous at every point of some dense G δ subset of X.
Proof. Every closed ball in L ∞ (µ) is a closed subset of L 1 (µ). We choose x 1 , x 2 , · · · ∈ X such that f (x k ) are L ∞ -dense in f (X). We consider closed balls B n,r in L ∞ (µ) of radius r centered at f (x n ). Their inverse images f −1 (B n,r ) are closed in X, and ∪ n f −1 (B n,r ) = X (for every r > 0). Denoting by U n,r the interior of f −1 (B n,r ) we observe that ∪ n U n,r is a dense open set in X and ∩ r ∪ n U n,r is a dense G δ set (Baire's theorem).
If y ∈ U n,r and y k → y then f (y) ∈ B n,r and f (y k ) ∈ B n,r for large k, therefore lim sup k f (y k )−f (y) ∞ ≤ 2r. So, if y ∈ ∩ r ∪ n U n,r and y k → y then lim sup k f (y k )− f (y) ∞ = 0.
Proof. The necessity is easy to see. The sufficiency follows from Lemma 2.5, applied to the map G → L ∞ (µ), g → f • g. By homogeneity, its continuity at a single point implies continuity everywhere.

Whirly actions
Often one can use the necessary and sufficient condition of Theorem 2.2 to verify directly that a given near action has no spatial model. This is done most easily by the following notion which will guarantee that a near action admits only constants as G-continuous functions.   V (gA, B) = gV (A, B), which shows that V (A, B) is dense in G.  Here is yet another equivalent definition. A near-action of G on (X, X, µ) is whirly, iff for every set A ∈ X of positive measure and every neighborhood U of e in G, µ(UA) = 1 ; here UA means ∪ n (g n A) where (g n ) is a dense sequence in U (its choice does not matter mod 0). Proof: µ((UA) ∩ B) > 0 ⇐⇒ ∃n µ((g n A) ∩ B) > 0 ⇐⇒ ∃g ∈ U µ((gA) ∩ B) > 0 ⇐⇒ ∃g ∈ U µ(A ∩ g −1 B) > 0.
We will next describe some applications of Proposition 3.3. Our first application will be to the natural near-action (on X) of the group G = Aut (X) of the automorphisms of the Lebesgue space (X, X, µ). We have already seen that this action has no spatial model since G is a Lévy group. There is however a more direct proof; we simply verify that the action is whirly. To this end recall that a neighborhood of the identity in G is given by a finite measurable partition of X into sets {P 1 , P 2 , . . . , P N } and ε > 0 as: For any sets A, B ∈ X of positive measure, if A 0 ⊂ A, B 0 ⊂ B are measurable, disjoint and have the same measure µ(A 0 ) = µ(B 0 ) < ε/2, and S is defined to be a measure preserving transformation which is the identity on X \ (A 0 ∪ B 0 ) and interchanges A 0 with B 0 , then S ∈ U and it satisfies µ(A ∩ SB) > 0.
The same kind of argument can be given for many subgroups of G and their natural near-action on X. For example we can start with any countable subgroup Γ ⊂ G that acts ergodically on X. The full group of this action [Γ], consists of all the measure preserving transformations T ∈ G = Aut (X, X, µ) such that for µ a.e. x ∈ X, T x ∈ Γx. Proof. The argument given above for the entire group G works here as well, almost verbatim. The only place where some change is needed is when we choose the transformation S; this time it should be in [Γ]. Now suppose we are given the open set U = U(P 0 , P 1 , . . . , P N ; ε) and two positive measure sets A, B ∈ X. If µ(A ∩ B) > 0 there is nothing left to show. Otherwise the ergodicity of the Γ-action guarantees the existence of a γ ∈ Γ with µ(A ∩ γB) > 0. Set A 1 = A ∩ γB and choose any A 0 ⊂ A 1 with 0 < µ(A 0 ) < ε/2. We let B 0 = γ −1 A 0 . The transformation S is now defined as the identity on X \ (A 0 ∪ B 0 ) and it interchanges A 0 with B 0 by means of γ and γ −1 . Clearly S ∈ [Γ], S ∈ U and it satisfies µ(A ∩ SB) > 0.
Remark 3.6. Let us note that, clearly, for every dense subgroup H of G = Aut (X, X, µ) the action of H on X is whirly. Moreover it can be shown that in the notation of the previous discussion, the group [Γ] is dense in G whenever Γ acts ergodically on X. However, proving the latter assertion requires a considerably more elaborate argument than the direct proof we provided in Proposition 3.5. In addition to the topology of convergence in measure on G = Aut (X, X, µ) one can consider the much stronger uniform topology given by the metric d u (S, T ) = µ{x ∈ X : Sx = T x}. Although with respect to this topology G is a non Polish topological group, the subgroup [Γ] is a closed Polish subgroup (see Hamachi-Osikawa [12], Lemma 53, or observe directly that for a countable generating collection of measurable partitions P = {P = (P 1 , . . . , P n )} of X and a fixed enumeration Γ = {γ 1 , γ 2 , . . . }, the set of elements {h(P ; k 1 , k 2 , . . . , k n ) ∈ [Γ] : P ∈ P, (k 1 , k 2 , . . . , k n ) ∈ N n }, where h(P ; k 1 , k 2 , . . . , k n ) ↾ P j = γ k j , j = 1, 2, . . . , n, is a countable dense subset of [Γ]). We now note that for both G and [Γ] our proofs show that their actions on (X, X, µ) are in fact whirly with respect to the uniform topology. Again we are indebted to V. Pestov for pointing this out.

The automorphism group of the Gaussian measure
We now turn back to the full orthogonal group G = O(l 2 ) acting on (R ∞ , γ ∞ ) as explained before Corollary 1.8. Thus we let ζ 1 , ζ 2 , · · · : X → R be i.i.d. N(0, 1) random variables defined as the coordinate functions on the space of sequences X = R ∞ equipped with its Borel σ-algebra X and the Gauss measure µ = γ ∞ . We identify l 2 with the closed linear subspace H ⊂ L 2 (µ) generated by the functions ζ 1 , ζ 2 , . . . ; namely, (c 1 , c 2 , . . . ) ∈ l 2 with c 1 ζ 1 + c 2 ζ 2 + · · · ∈ H. The near-action is given by  The following technical lemma is important for the proof. Basically it states that a remote perturbation of a finite-dimensional condition forces the conditional probability to be strictly positive. Note that the relation P A . . . > 0 may be written as sgn P A . . . = 1, using the (discontinuous) sign function.

Proof. (a)
We introduce functions f n : R → [0, 1], g n : R 2 → [0, 1] by P A ζ 1 cos α + ζ n sin α = f n (ζ 1 cos α + ζ n sin α) , and a set B ⊂ R by . (These f n , g n , B are treated mod 0, of course.) We have P A ζ 1 , ζ n → P A ζ 1 in probability (for n → ∞). On B × R we get sgn g n → 1 in measure, with respect to γ × γ, where γ = N(0, 1) is the one-dimensional Gaussian measure. However, any equivalent (that is, mutually absolutely continuous) finite measure on B × R may be used equally well.
Taking into account that f n results from g n by integration (along straight lines orthogonal to the unit vector (cos α, sin α)) we get sgn f n (u) ≥ ess sup x sgn g n x, u − x cos α sin α .
The map (x, u) → x, u−x cos α sin α of B × R to itself sends the measure to an equivalent measure. So, sgn g n x, u−x cos α sin α → 1 in measure, which implies sgn f n → 1 in measure (with respect to γ).
Proof of Theorem 4.1. Let A ∈ X be a set of positive measure, and U a neighborhood of e in G; by Remark 3.4 it is sufficient to prove that µ(UA) = 1. Of course, UA is treated as in Remark 3.4 (and the same about ZA for any Z ⊂ G).
Appendix A.
Measure concentration (that is, the property of being a Lévy family or group) is proven for various cases by a number of methods [23], [16]. Strong results need complicated proofs involving advanced methods (Riemann geometry, representation theory, etc.). More elementary arguments give weaker results which are satisfactory for many topological applications such as the ones we needed in Section 1. This appendix collects elementary (complete) proofs for many Lévy families.
A.2. Euclidean spheres S n−1 = {x ∈ R n : |x| = 1} are a Lévy family, since a random point of S n−1 can be obtained from a Gaussian random vector ξ ∈ R n distributed γ n

It remains to apply A.3.
A.5. The commutative Polish group L 0 ([0, 1], S 1 ) is a Lévy group. It consists of all equivalence classes of measurable functions [0, 1] → S 1 , where S 1 = {z ∈ C : |z| = 1}, and is in fact monothetic [7]. The following proof of its Lévy property is basically an extract from [16,Sect. 1.6,4.1]. See also [16, p. 31] and [7]. Let G be a commutative Polish group with a compatible invariant metric d, and µ a Borel probability measure on G; we define Clearly, (the diameter of the support). It is easy to see that , and the squares are taken pointwise. For any two measures µ, ν The argument is applied to G = L 0 ((0, 1), S 1 ) as follows. We choose the L 1 -metric For each n we consider the n-dimensional compact subgroup K n = K n,1 + · · · + K n,n ⊂ G where K n,m is the one-dimensional group of functions constant on ( m−1 n , m n ) and equal to 1 on (0, 1) \ ( m−1 n , m n ). The corresponding invariant measures are related by m Kn = m K n,1 * · · · * m Kn,n . However, therefore VarLip 2 (m Kn ) ≤ n · ( 2 n ) 2 → 0 for n → ∞. It remains to use the relation between Lévy families and Lipschitz functions mentioned in A.1.
We apply the argument to the group S n of all permutations of {1, . . . , n} equipped with the Hamming metric d(g, h) = #{k : g(k) = h(k)} n .
Its invariant measure m Sn is the convolution of n measures, each concentrated on transpositions (that is, g such that d(g, e) ≤ 2/n). Indeed, S n−1 is naturally embedded into S n , and m Sn = m S n−1 * µ where µ is distributed uniformly on transpositions of n and k for k = 1, . . . , n. So, VarLip 2 (m Sn ) ≤ n · ( 4 n ) 2 → 0 for n → ∞. It remains to note that there exists a natural embedding of the inductive limit group S = lim n→∞ S 2 n as a dense subgroup of G = Aut ([0, 1]). Here the group S 2 n is embedded into S 2 n+1 as the subgroup of permutationsσ of {0, 1, . . . , 2 n+1 − 1} of the formσ(2k) = 2σ(k) andσ(2k + 1) = 2σ(k) + 1, k = 0, 1, . . . , 2 n − 1, σ ∈ S 2 n . Then, for each n, the group S 2 n is identified as the subgroup of G = Aut ([0, 1]) which consists of the transformations permuting the 2 n dyadic sub-intervals of [0, 1] by translations. It can be easily seen that the restriction of the uniform metric d u (S, T ) = µ{x ∈ X : Sx = T x} on G to S 2 n is the Hamming metric. Thus with respect to this metric and using the estimation for VarLip 2 (m Sn ) we see that S is a Lévy group. Since the identity map from (S, d u ) to G is continuous and since S is dense in G we can finally conclude that also G is a Lévy group. See [9] and also [16,Corollary 4.3].
Appendix B.
The theorem stated below is well known and widely used. However it seems that complete proofs are not easily found. The proof we provide is from Oxtoby [25] where it is attributed to Ulam. Proof. Let µ be a σ-finite measure on G such that µ(B) > 0 ⇐⇒ µ(gB) > 0, ∀ measurable B, ∀g ∈ G.
Let A be a measurable subset of G with 0 < µ(A) < ∞. Since every Borel measure on a Polish space is regular there exists a compact set K ⊂ A with 0 < µ(K) < ∞.
Let H < G be the subgroup of G which is generated by L = K ∪ K −1 . Clearly H = ∪{L n : n ∈ N} is a σ-compact group. If G/H is uncountable then there are uncountably many distinct cosets of H in G and in particular uncountably many pairwise disjoint translations of K. This however contradicts the σ-finiteness of µ and we conclude that G/H is countable. Now Baire's theorem implies that int L n = ∅ for some n and we conclude that G is locally compact.
Remark B.2. In [18, theorem 7.1] Mackey proved a more general theorem. He showed that if G is an analytic Borel group (no topology is given) which admits an invariant σ-finite measure class then there exists a unique locally compact topology on G whose Borel structure is the given one and under which G is a topological group. His proof relies on Weil's theorem, [33]. Finally we note that in [1] A. D. Alexandroff proved a related result (he is not assuming σ-finiteness of the invariant measure but, instead, the existence of an open set with finite positive measure).