Restoring Topology from Shifts
Abstract
Description
It is known that the topology of a Polish group is uniquely determined by its Borel structure and group operations, but this does not give us a way to find the topology. In this article we expand on this theorem and give a criterion for a measurable function on the group to be continuous. We show that it is continuous iff there exists some second countable topology in which all the shifts of the function are continuous. Here measurability can be taken to be measurable with respect to Haar measure (in locally compact groups), or having the Baire property or universally measurable (in Abelian Polish groups). Our results appear to be new even when the group is $\R$. As a special case we get that a measurable homomorphism is continuous (a known result).
As an application, We give a first proof of a dichotomy guessed by Tsirelson long ago for stationary stochastic processes. Either the process is sample continuous, or its paths cannot be continuous in any second countable topology on any non-null set.
24 pages
24 pages