Errata: Dynamic Programming, Successive Approximations, and Monotone Convergence
Proceedings of the National Academy of Sciences of the United States of America
the triple (P( 1A), AF, A) where A = U jA3 = U j [w; x(a(w), w) = j] will follow using our proof if one chooses the approximating an(w) more delicately. Let R5 = [2-'; m = 1, 2 ... ] for p = 1, 2 ... and let R = U ,Rp be a separability set for x(t, w). Since x(t, w) e 9t* the set [uw; w e A1, ac(w) <t] e T, Ix(s), s < t . Let xr,[O = a" , < a2, , < ... ] be a sequence of partitions of [0, co) with norms tending to 0 and define the optional random variables as ap*(w) = ak on Bk = [W; ak-i < a <
... k = [W; ak-i < a < ak] with left inclusion at k = 1. Fix integers j, k and p and let Ai = Ai(k = [w; w e Ai n Bk, a(w) < Si, X(S, w)where si < 82 . . . < sq is8 an ordering of Rp n (ak-i", ak,5]. We define a,(w) = sI on AI -U A1, 1 = 1, 2 ... q; the r.v.'s are defined similarly 1=1 for each k and j and we let a,(w) = a"*(w) elsewhere. The r.v.'s a"(w) are optional and satisfy a"(w) I a(w) and x(a", w) --x(a, w) on A; hence we may let tLi = t* = 0 on the triple (P( -1A), AF, A) in the proof of our theorem and obtain the stronger result. Finally we mention that Jushkevich has quite recently (Russian J. of Prob.) obtained results on the strong Markov property in a different form from that presented here. Also Chung has employed the continuity of the conditional distribution of y(t) relative to a to show that y(t) is separable without modification.