H; if there exists a sequence (H) announcing H, we say H is n announceable (foretellable). In the one-parameter theory, this is equivalent to being predictable (and in fact many authors define predictability in this fashion). However, we only have one implication for stopping lines, namely, that every announceable stopping line is predictable. In fact, if H is announceable, then ex. (H ) such that [H,-) ( ,n' nn Each (Hn,.) is predictable, hence the intersection is predictable. By remark (4), this implies that H is predictable. Unfortunately, the other implication does not hold. For a counterexample, see Bakry l[]. Finally, we note the existence of a predictable cross-section theorem for stopping lines. The proof is essentially the same as for the one-dimensional case. We denote by i the projection of R x onto a. Theorem 1.4.4 (predictable cross-section theorem). Let A be a predictable set, and let e>0. There exists a compact* predictable set K satisfying the following: 1) KC A and P{K = 0, A 0} < E 2) DK is announceable and K C D. 1.5 Some Measure Theory In this section we collect some results from measure theory which we shall make frequent use of later. * That is, the section K(w) is compact for each w.