  We initiate a systematic study of the convolution operation on Keisler
measures, generalizing the work of Newelski in the case of types. Adapting
results of Glicksberg, we show that the supports of generically stable (or just
definable, assuming NIP) measures are nice semigroups, and classify idempotent
measures in stable groups as invariant measures on type-definable subgroups. We
establish left-continuity of the convolution map in NIP theories, and use it to
show that the convolution semigroup on finitely satisfiable measures is
isomorphic to a particular Ellis semigroup in this context.
