On Filippov’s implicit functions lemma

In 1959 A. F. Filippov published a paper [1] containing a lemma designed for use in the study of optimal control problems. Stated somewhat imprecisely, let k be a continuous function on a compact set Q in a finite dimensional space and with values in a finite dimensional space, and let (u'(t): a?<t <b) be a function with values in Q such that k(u'(.)) is measurable; then there exists a measurable function u from [a, b] to Q such that k(u(t)) ==k(u'(t)). For purposes of stochastic control theory it is desirable to extend this to allow arbitrary measure spaces, instead of intervals of real numbers (Kushner [4]); for purposes of the calculus of variations it is desirable to relax the requirement of compactness on Q. We do both of these (Q may be any separable metric space), and at the same time we permit the values of k to lie in any Hausdorff space; it costs nothing. In ?2 we give an application to optimal control theory. ADDED IN PROOF. In an abstract in Amer. Math. Monthly 73 (1966), p. 927, M. Q. Jacobs announces a generalization of Filippov's lemma. This generalization is a special case of our Theorem 1.