On limiting behavior of probability measures on locally compact semigroups (Q1806243)

Let \(S\) be a locally compact, Hausdorff, second countable semigroup. Let \(\mu\) be a probability measure on \(S\) such that the sequence of \(n\)-times convolution powers \(\{\mu^{*n}\), \(n\in{\mathbb N}\}\) is tight. Let \(A=(a_{nj})\) be an infinite Toeplitz matrix (i.e. which transforms convergent sequences into convergent sequences and which preserves limits) satisfying a strong regularity condition (i.e. a normal continuity assumption, on the lines, at infinity). The authors prove that the generalized Cesaro sum \(\sum a_{nj}\mu_j\) converges weakly to an idempotent probability measure which is independent of \(A\). The result is also true when \(S\) is a Polish space, and an analogous result holds true for the vague topology without the tightness assumption. This apparently generalizes a theorem of \textit{J. Liu} and \textit{K. Xu} [in: Semigroup theory and its related fields, 129-136 (1990; Zbl 0746.60010)].