Existence of Regular Finite Invariant Measures for Markov Processes

F. Dennis Sentilles
1969 Proceedings of the American Mathematical Society  
Let S be a locally compact Hausdorff space and let P(t, x, •) be a transition function on the Borel subsets of 5 with P(t, x, S)^a>0 for all ¿S:0, xES. In a recent paper V. E. Benes [l] obtains several necessary and sufficient conditions that P(t, x, ■) have an invariant measure ju in the space M(S)+ of strictly positive bounded regular Borel measures on 5 in the presence of several overriding conditions on the function P and the space 5. We propose to eliminate the need for these conditions by
more » ... making use of the fact that M(S) is the dual of the locally convex space C(S) oí bounded continuous functions on 5 with the strict topology of Buck [3] (see also [9] for a more general discussion of this topology) and replace the use of weak compactness in M(S) in [l ] by that of /3-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(5))-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 / ^ 0 the function [Ttf ] (x)
doi:10.2307/2036992 fatcat:nuisgw7scne6hoiwrw34yfvjlq