Change of time scale for Markov processes

for all AEG. Let Q be an invariant measure. We assume the following conditions: (i) S is a locally compact topological space, (B is generated by the compact sets, and Q is a regular measure. (ii) (3 is separable, SC (B, Q(S) > 0. (iii) Q is a sigma-finite invariant measure such that P [entering A at some time I Xo = x] = 1 for all xCS and all A C (B with Q(A) > 0, where P is the underlying probability measure. The conditional probability in (iii) is to be taken as determined by the transition probability function. The same remark applies to all similar situations appearing subsequently. Processes satisfying (ii) and (iii) were introduced by Harris in [5] and studied further in [7](2). Here we shall use the word recurrent for processes satisfying (i)-(iii). We continue to use Q for some measure satisfying these conditions; in any case, according to [5] such a measure is unique up to constant multiple. Let y(x) be a measurable function from (S, (3) into the positive reals satisfying