The sets of constancy

Abstract

From now on we assume that the minimal state space I of the M. C. {x _t , t ∈ T} is discrete and that it is compactified to $\bar{I}$ by adjoining the fictitious state ∞. We also assume that the transition matrix (p _ij )is standard. Instead of going at once to a well-separable and measurable version as guaranteed by Theorem 4.3, we shall first investigate separately the consequences of separability and measurability. In this and the next two sections a theorem which is proved under the further assumption of separability is marked with (S), that of measurability with (M), that of both with (SM).