  We study idempotent measures and the structure of the convolution semigroups
of measures over definable groups. We isolate the property of generic
transitivity and demonstrate that it is sufficient (and necessary) to develop
stable group theory localizing on a generically stable type, including
invariant stratified ranks and connected components. We establish generic
transitivity of generically stable idempotent types in important new cases,
including abelian groups in arbitrary theories and arbitrary groups in rosy
theories, and characterize them as generics of connected type-definable
subgroups. Using tools from Keisler's randomization theory, we generalize some
of these results from types to generically stable Keisler measures, and
classify idempotent generically stable measures in abelian groups as (unique)
translation-invariant measures on type-definable fsg subgroups. This provides a
partial definable counterpart to the classical work of Rudin, Cohen and Pym for
locally compact topological groups. Finally, we provide an explicit
construction of a minimal left ideal in the convolution semigroup of measures
for an arbitrary countable NIP group, from a minimal left ideal in the
corresponding semigroup on types and a canonical measure constructed on its
ideal subgroup. In order to achieve it, we in particular prove the revised
Ellis group conjecture of Newelski for countable NIP groups.
