Suppose {F t } is a filtration satisfying the usual conditions. Show that if S and T are stopping times and X is a bounded F∞ measurable random variable, then   A martingale or submartingale Mt is...

Suppose {F_t} is a filtration satisfying the usual conditions. Show that if S and T are stopping times and X is a bounded F∞ measurable random variable, then

A martingale or submartingale Mt is uniformly integrable if the family {M_t
: t ≥ 0}is a uniformly integrable family of random variables. Show that if Mt is a uniformly integrable martingale with paths that are right continuous with left limits, then {MT ; T a finite stopping time} is a uniformly integrable family of random variables. Show this also holds if Mt is a non-negative submartingale with paths that are right continuous with left limits.