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We extend the results of Csiszar (Z. Wahr. 5(1966) 279-295) to a topological semigroup S. Let μ be a measure defined on S. We consider the value of α=supKcompactlimn→∞supx∈Sμn(Kx−1). First. we show that the value of α is either zero or one. If α=1, we show that there exists a sequence of elements {an} In S such that μn∗δan converges vaguely to a probability measure where δ denotes point mass. In particular, we apply the results to inverse and matrix semigroups.

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International Journal of Mathematics and Mathematical Sciences, v. 8, art. 135913