Universally measurable subgroups of countable index

TL;DR

The paper proves that universally measurable subgroups of countable index in Polish groups are open, leading to the continuity of certain homomorphisms into symmetric and locally compact groups.

Contribution

It establishes that universally measurable subgroups of countable index in Polish groups are open, and that homomorphisms into symmetric or locally compact groups are continuous.

Findings

Countable index, universally measurable subgroups are open in Polish groups.

Universally measurable homomorphisms into $S_inite$ are continuous.

Homomorphisms into second countable, locally compact groups are continuous.

Abstract

It is proved 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 $S_\infty$ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.