Show description

For each n, F„ is probability measure on ^„ and P„+i agrees with P„ on J^„. Hence there is a unique finitely additive P on Q„ ^„ such that P agrees with P„ on ^ „ . We will show that P is countably additive on |J„ J^„ and therefore extends uniquely as a probabihty measure to J^. We have to show that if B„ e ij„ J^„ and J5„i0 then P(B„)iO. Assume that P(B„) > e > 0 for all n. We will produce^a point qef]„B„. We can assume, without loss of generality, that B „ G J ^ „ . For 0 < M < m and B G J^„ we define TT'"- "(q, B) to be XB(Q)- For n > m and B e J^„ we define TT'"' "(q, B) inductively by n"''"{q,B)= \n"{q',B)n'"'"-'(q,dq).

1) Jf = (7[x(t): r > 0 ] . 2) ^ , = (7[x(s): 0 < 5 < r]. 3. 3) M, = G[\]JI\ for t > 0. 4) ^ = (7|U^^r). 4. 1 Theorem. 8) lim sup ^iO (ae K 0 sup \x(s, oj) — x(t, oj)\ =0. 3. We now prove the sufficiency. 5) such that infP[|x(0)| < ^ ] > 1 P e^ If we define K. = r\ (jo\ sup 0 1 — £ for all P e ^. Moreover by the Ascoli-Arzela theorem Kg is compact in Q.

12), respectively, we know that E[d(x„%e(x„)>x]k] and £[0(ff„), 0(c„) > A] < £[^(T), 0(ff„) > X]. 4), p( sup0(t)>A)<|£[e(T)], \o 1} and {^(T„): « > 1} are uniformly Pintegrable. Since 9 is P-almost surely right continuous, we now know that 6((7„) -• 6((7) and 9(x„) -• 9(x) in L^(P). 8). 26 1. Preliminary Material: Extension Theorems, Martingales, and Compactness The proof that {^(T): T a stopping time bounded by 7} is uniformly integrable when (9(t), J^,, P) is a non-negative submartingale is accomphshed in exactly the same way as we just proved that {^(T„): W > 1} is uniformly integrable.