Pointwise in terms of weak convergence

Abstract:

Let $(\Omega ,\mathfrak {F},\mu )$ be a measure space, $\mu (\Omega ) < \infty$. Let ${X_n}$ be a sequence of measurable functions on $\Omega$ taking values in a compact metric space $M$. The set of bounded stopping times $\tau$ for the ${X_n}$ is a directed set under the obvious ordering. The following theorem is proved: ${X_n}$ converges pointwise almost everywhere if and only if the generalized sequence $\int {\phi ({X_\tau })d\mu }$ converges for every continuous function $\phi$ on $M$. The martingale theorem is proved as an application.