Amenability, free subgroups, and Haar null sets in non-locally compact groups

The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are 'amenable at the identity', and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable groups.