Polish topologies on groups of non-singular transformations

TL;DR

This paper investigates the topological properties of groups of non-singular transformations, establishing the uniqueness of the natural Polish topology and conditions for automatic continuity.

Contribution

It proves the non-existence of Polish group topologies on certain transformation groups and characterizes measures for automatic continuity.

Findings

No Polish topology on measure-preserving transformations with finite support

Characterization of measures with automatic continuity property

Uniqueness of the natural Polish topology on non-singular transformations

Abstract

In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel $\sigma$-finite measures $\lambda$ on a standard Borel space for which the group of $\lambda$-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.