Equivariant concentration in topological groups

In this chapter we introduce several most important notions and constructions concerning topological groups and operations with them. In particular, we develop the technique of prenorms on groups, describe in detail the construction of the Raĭkov completion of a topological group, prove a theorem on embeddings of topological groups into groups of isometries in the topology of pointwise convergence. We also introduce the class of locally compact topological groups and provide the reader with the first basic facts about groups in this class (deeper results in this direction will be presented in Chapter 9). Then we define and study the important classes of !-narrowtopological groups and of precompact (equivalently, totally bounded) topological groups. We finish this chapter with the Hartman{Mycielski construction of an embedding of an arbitrary topological group into a connected, locally pathwise connected topological group.

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.

Some aspects of dimension theory for topological groups

Let G be an arbitrary topological group. We prove that the cellularity of G is equal to the cellularity of the quotient space G / K {G/K} for every compact subgroup K of G.

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.

Fourier–Stieltjes algebra of a topological group

In the present paper, the continuous Hu cohomology with coefficients is defined in an arbitrary topological abelian group. The aim of this work is to find the connection between this cohomology and the Alexander–Spanier cohomology for different groups of coefficients.

We state a simple fact relating some continuity and homomorphy properties of an internal mapping between nonstandard extensions of topological groups and the nonstandard extension of the “observable trace” of the map, which can be interpreted as a kind of stability principle. This leads to a strengthening of two formerly proved (standard) stability results (a global one and a local one) along with simplifying their proofs. We show that every “sufficiently continuous,” “reasonably bounded” and “sufficiently homomorphic” mapping from a locally compact to an arbitrary topological group is “arbitrarily close” to a continuous homomorphism between them.

In this paper, we primarily introduce the concepts of random measure-theoretic N-expansivity and random measure-theoretic cw-expansivity for a Borel probability measure, as well as the notions of random N-expansivity and random cw-expansivity for a random dynamical system. Subsequently, we prove that a random dynamical system is randomly N-expansive (res., randomly cw-expansive) if and only if every Borel probability measure is randomly measure-theoretic N-expansive (res., randomly measure-theoretic cw-expansive) of f and all Borel probability measures share a common random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic), namely, there is a measure random variable δ : Ω → ( 0 , + ∞ ) such that δ is a random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic) of every Borel probability measure.

We consider Hyers-Ulam stability of a functional equation for continuous functions on a space on which a topological group acts, analogous to the additive functional equation on a group. We show, among other things, that our generalized additive equation, for continuous functions on a homogenous space of a strongly amenable topological group, is stable provided that the canonical projection from that group to its homogenous space is a fiber bundle.

Admissible Representations of Probability Measures

Continuum-wise expansive measures

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H.As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.In [9], G. W. Mackey gave an argument using Haar measures to prove the following: Assume G is locally compact topological group.If H is a subgroup such that the space G/H, formed by giving the set of (left) H-cosets the quotient Borel structure, is countably separated, then H is closed in G.We will show that this result can be obtained by an analogous argument using the theory of Baire category.The category version has a wider application and shows that the above statement remains true under the weaker assumption that G is totally nonmeager.In particular, it holds whenever G is topologically complete (in the sense of Cech).As a corollary, we will prove that a well-known theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish (separable, completely metrizable) groups holds for Borel actions as well.We will also show that a recent result of R. Vaught on decompositions of invariant analytic sets has a similar extension to Borel actions.Each step in our main argument corresponds to an analogous step in Mackey's proof.The reader may find it interesting to compare our development with the proof of Theorem 7.2 in Mackey [9].Recall (cf.[2, IX,) that a topological group G which is not meager (first category) is a Baire space (i.e.no nonempty open subset of G is meager in G).G is totally nonmeager if no nonempty closed subset of G is meager in itself.Every topologically complete space, in particular every locally compact space and every completely metrizable space, is totally nonmeager (cf.[7, III, 8]).B E G is strictly Baire (or has the restricted Baire property)