On measurable stochastic processes

Warren Ambrose
1940 Transactions of the American Mathematical Society  
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
more » ... erned 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.
doi:10.1090/s0002-9947-1940-0000918-4 fatcat:kd26fdt5kjb6rpp6lx2x2tqfi4