Markoff processes and potentials I

This paper will appear in three parts.Its purpose and scope are best dis- cussed after the state of affairs in the simplest situation has been explained.Let P(r, ds) be stationary Markoff transition measures on 5C, a separable locally compact space, and suppose that the transformations Pr of functions on , Pr f.(r) J P(r, ds)f(s), leave invariant the Banach space of continuous functions vanishing at in- finity and converge strongly there to the identity transformation as r--0.The transition measures can be realized by Markoff processes whose sample paths are continuous on the right and have limits from the left; the letter X will denote such a process, X(r) the random point of obtained by fixing the time as r, and X(r, ) the point of 5C obtained by fixing in addition the element of the basic probability space over which the process is defined.A set E in 5C is said to be nearly Borel if, for each process X, there are Borel sets A and B such that A E B and such that, for almost all o and for all r, the point X(r, 0) belongs to all three sets if it belongs to one.Such sets form a Borel field; a function measurable over the field is said to be nearly Borel measurable.The time T at which a process X hits a nearly Borel set E is defined to be the infimum of the strictly positive r for which X(r, o) belongs to E, or if there are no such r; it is a random variable, that is to say, a measurable function on the basic probability space.If X starts at a point, T vanishes with probability either 0 or 1.In the latter event the point is said to be regular for E; for example, an interior point of E is certainly regular for E. A point r and a nearly Borel set E determine a measure H(r, ds), defined by the formula H(r, A) {X(T(), ) e A, T(0) < }, where X is a process starting at r and (P/" stands for the probability of the event within the curly brackets.A few remarks on language and notation are needed.Let a be the Borel field comprising the sets in 3C that are measurable for the completion of every measure defined on the topological Borel field of 3C.A function on 3C is understood to be measurable over a; a positive function may take on the values 0 and a measure on 5C is understood to be defined on a and to be the sum of countably many bounded positive measures.A kernel, say