Pointwise in terms of weak convergence

Let (Ω,?, p) be a measure space, µ(Ω) ∞. Let X be a sequence of measurable functions on Ω taking values in a compact metric space M. The set of bounded shopping times r for the X isa directed set under the obvious ordering. The following theorem is proved: X converges pointwise almost everywhere if and only if the generalized n sequence 4Ø(Xt)du converges for every continuous function Ø on M. The martingale theorem is proved as an application.