On limiting behavior of probability measures on locally compact semigroups

In this paper we study the problem of summability in the weak and the vague topology of the sequence of convolution iterates (mu(n), n is an element o f N) where mu is a probability measure on locally compact semigroup S. Inst ead of the usual Cesaro, (c, 1) summability of a sequence (mu(n), n is an e lement of N), we consider the problem of the summability of this sequence b y stochastic strongly regular matrices. Our Theorem 1 generalizes Theorem 3 .1 in [6, p. 131]. Theorem 1 also holds if we make some changes on the topo logical assumptions on S, i.e., if we suppose that S is a Polish space.