This led to many applications, e.g. resolution of Pillay’s conjecture for compact o-minimal groups, or Hrushovski’s work on approximate subgroups. And brought to light the study of invariant measures on definable subsets of the group, and the methods of topological dynamics.

In this paper we obtain a remarkable counterpart of the aforementioned classical theory for locally compact topological groups, and classify idempotent generically stable measures in abelian groups as (unique) translation invariant measures on type-definable subgroups.