  This expository paper is intended for a short self-contained introduction to
the theory of infinite convolutions of probability measures on Polish
semigroups. We give the proofs of the Rees decomposition theorem of completely
simple semigroups, the Ellis--\.{Z}elazko theorem, the convolution
factorization theorem of convolution idempotents, and the convolution
factorization theorem of cluster points of infinite convolutions.
