Existence of regular finite invariant measures for Markov processes.

Abstract

Let S be a locally compact Hausdorff space and let P(t, x, *) be a transition function on the Borel subsets of S with P(t, x, S) O for all t_O, xCS. In a recent paper V. E. Benes [1] obtains several necessary and sufficient conditions that P(t, x, *) have an invariant measure , in the space M(S)+ of strictly positive bounded regular Borel measures on S in the presence of several overriding conditions on the function P and the space S. We propose to eliminate the need for these conditions by making use of the fact that M(S) is the dual of the locally convex space C(S) of bounded continuous functions on S with the strict topology of Buck [3 ] (see also [9] for a mnore general discussion of this topology) and replace the use of weak compactness in M(S) in [1] by that of :-weak* compactness studied by Conway [4] (these two notions of compactness are studied in [8]) or equivalently, in the notation of [6, p. 32], a(M(S), C(S))-compactness. The referee has pointed out that our results are also an improvement of some of the results in [2 ], where the same author does replace weak compactness in M(S) by a(M(S), Co(S))-compactness in the presence of certain other conditions. After proving our main theorem we will discuss [2 ] in more detail. Our only assumption on P is that for each t ? Othe function [Ttf] (x) fsf(s)P(t, x, ds) belongs to C(S) for all fEC(S). Equivalently, the function x->P(t, x, *) is a continuous function on S into M(S) with the a(M(S), C(S)) =a(M, C) topology, for in this topology a net v,--+v if and only if fsfdva,->fsfdv = (f, v) for all f E C(S). It follows from [71 that for each .t EM(S) and Borel set E the formula