On Modification Theorems

Given a right continuous family Ft of complete o-fields and a bounded right continuous family Xt of random variables, we show in this paper that it is possible to modify the conditional expectations E(Xt\Ft) to be right continuous. When Xt=X, this reduces to a result of J. L. Doob. A famous result of Doob states that any martingale has a right continuous with left limits modification. This is an important result and is very useful in providing modification theorems in the theory of Markov processes. During a discussion with Professor T. Watanabe the following problem arose: Suppose Ft is an increasing right continuous family of a-fields and Xt is a right continuous with left limits stochastic process. Can we define a right continuous with left limits modification of E(Xt\Ft)l We shall show that it is always possible to select such a modification. Note that whereas in the case of martingales we have available the upcrossing inequality of Doob (this is an indispensable tool in the standard proofs of the martingale modification theorem) there is no such tool in the general situation. The method developed here generalizes without changes to Banach space valued martingales. Thus a Banach space valued martingale has a right continuous with left limits modification. Our methods do not require the notion of separability. The only prerequisite to reading this note is the knowledge of the martingale convergence theorem and familiarity with the notion of stopping rules. In the beginning of §1 we develop what is needed about stopping rules in a more general setting than what is given in standard books. Then we proceed to prove the main theorem (Theorem 6). In §2 we show how the modification theorem for super martingales can be deduced using Theorem 6. Lemma 9 gives a slight generalization of a theorem of Meyer without invoking the upcrossing inequality. Thanks are due to Karl Pedersen for many valuable discussions. 1. Let F„ denote a sequence of a-fields for «= 1, 2,..., oo; oo is included. An integral valued nonnegative function Fis called a stopping rule relative to Fn iff for all « the event (T=n) e Fn. If Fis a stopping rule we denote by FT the a-fields of events A such that A n (T=n) e Fn, 1 ¿«¿oo. It is easy to verify that FT is indeed a CT-field. We simply note the following: 1. T is Fr-measurable. Received by the editors November 11, 1970. A MS 1970 subject classifications. Primary 60GXX.