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.