On Measurable Stochastic Processes

In recent years probability theory has been formulated mathematically as measure theory; in the case of stochastic processes depending upon a continuous parameter the measures considered are defined on certain subspaces of the space of all functions of a real variable.! This formulation of stochastic processes depending upon a continuous parameter gives rise to certain measurability problems, and it is with these measurability problems that this paper is concerned. In particular we shall be concerned with conditions under which there will exist what Doob has called a measurable stochastic process.{ In §1 we give the necessary mathematical formulation of the notion of a stochastic process. In §2 we obtain general conditions for the existence of a measurable process, while in §3 we use the results of §2 to obtain conditions upon the conditional probability functions which are necessary and sufficient for the existence of a measurable process. In §4 we prove a theorem which is essentially due to W. Doeblin concerning the existence of a special sort of measurable process in case the conditional probabilities satisfy certain regularity conditions. 1. Mathematical formulations. We shall denote by ß the space of all realvalued functions of a real variable. We introduce a topology on ß by defining neighborhoods as follows: if h, • • • , t« is any finite set of real numbers and if ai, • • ■ , an and bi, ■ ■ ■, bn are sets of real numbers satisfying — °o ;S cti <bt s| « (i = \, ■ ■ ■ , n), then the set of elements x(t) of ft which satisfy ai<x(tl) <bi ■■•,») is a neighborhood. Next we consider a probability measure P(M), defined on the Borel field of sets determined by the collection of neighborhoods;! we shall suppose the domain of definition of the measure P(M) to be so extended that if P(M)=0 for a certain set M, then P(N) is defined for every subset N of M. The sets for which P(M) is defined will be called P-measurable. If N is any set in fl, we define its outer P-meas-