Universally measurable subgroups of countable index

Abstract

We prove that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S∞ is continuous. We also show that a universally measure homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous. The present work is motivated by an old problem of J.P.R. Christensen, which asks whether any universally measurable homomorphism between Polish groups is continuous. To fix the terminology, let us recall that a Polish space is a separable topological space whose topology can be induced by a complete metric. Also, a topological group is Polish in case its topology is Polish. A subset A of a Polish space X is said to be Borel if it belongs to the σ-algebra generated by the open sets and A is universally measurable if it is measurable with respect to any Borel probability (or equivalently, σ-finite) measure on X , i.e., if for any Borel probability measure μ on X , A differs from a Borel set by a set of μ-measure 0. The class of universally measurable sets clearly forms a σ-algebra, but contrary to the class of Borel sets, there is no “algebraic” procedure for generating the universally measurable sets from the open sets. Thus, the extent of this class is somewhat elusive and largely depends on additional set theoretical axioms. A function f : X → Y between Polish spaces X and Y is said to be Borel measurable, resp. universally measurable, if f−1(U) is Borel, resp. universally measurable, in X for all open U ⊆ Y . A classical result due to H. Steinhaus and A. Weil states that if G is a second countable, locally compact group (and hence Polish) and H is a Polish group, then any universally measurable homomorphism π : G → H is continuous. Actually, for this it suffices that π is measurable with respect to left or right Haar measure on G. However, this result relies heavily on the translation invariance of Haar measure, and, as any Polish group with a non-zero, quasi-invariant, σ-finite Borel measure is necessarily locally compact, the proof gives no indication of whether the result should hold for general PolishG. Nevertheless, in the late 1960’s, J.P.R. Christensen [1] (see also [2]) introduced a notion of Haar null sets in more general Polish groups and was able to use this to prove an analogue of the Steinhaus–Weil result for Abelian Polish groups G. In fact, a consequence of Christensen’s proof is that any universally measurable homomorphism from a Polish groupG into a Polish groupH , whereH admits a compatible two-sided invariant metric, is continuous. In particular, this applies to the case whenH is Abelian, compact or countable discrete. It immediately follows from this that if G is Polish and N 6 G is a countable index, universally measurable, normal subgroup, then N is open in G. To see this, one just considers the quotient mapping π : G → G/N , where G/N is taken discrete. Recently, Date: November 2008. 2000 Mathematics Subject Classification. 03E15.

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Cite this paper

@article{Rosendal2010UniversallyMS, title={Universally measurable subgroups of countable index}, author={Christian Rosendal}, journal={J. Symb. Log.}, year={2010}, volume={75}, pages={1081-1086} }