Existence of invariant measures for Markov processes. II

Abstract

1. Notation. We shall use the notation of [1] for topological and measure theoretic concepts. Let X be a normal topological space. Let P(x, A) be the transition probabilities of a Markov Process: 1.1. For a fixed xGX the set function P(x,) is a measure, on the Borel sets, of total measure one. 1.2. For a fixed Borel set A, the function P(., A) is Borel measurable. By a measure we shall mean a countably additive positive measure, unless otherwise stated. Let us denote by r b a the set of regular bounded finitely additive signed measures on X and by r c a those elements of r b a which are countably additive. The transition probabilities induce an operator on the bounded measurable functions by 1.3. (Pf) (x) =ff(y)P(x, dy). Also if j. is a bounded finitely additive signed measure one defines 1.4. (juP) (A) =fP(x, A),ju(dx). It is well known that 1.5. f(Pf) (x) A (dx) =ff(x) (juP) (dx) and that ,uP is countably additive if A is. Throughout the paper we assume: 1.6. If fE C(X) then PfE C(X), where C(X) denotes the continuous functions. Also: 1.7. If .lEr ca then APCrc a. These two conditions are always satisfied under the assumptions of [2 ]: under the assumptions of [2 ] every countably additive measure is regular. Another example is given by (PD (x) =f(?>(x)) where 4, is a homeomorphism of X onto X.