### Polish topologies on groups of 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.

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.

#### Keywords

Polish group, Non-singular transformation; infinite measure-preserving transformation; Automatic continuity

#### Full Text:

4. [PDF]DOI: https://doi.org/10.4115/jla.2022.14.4

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Journal of Logic and Analysis ISSN: 1759-9008