Universally measurable subgroups of countable index

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 -algebra generated by the open sets and A is universally measurable if it is measurable with respect to any Borel probability (or equivalently, -finite) measure on X, i.e., if for an...

• Type: Journal
• Year: 2010
• Where: JSYML
• Authors: Christian Rosendal
• Keywords: JSYML 2010, Measurable Homomorphism, Polish Group, Universally Measurable