  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.
