Convergence theorems of operator-valued martingale indexed by directed set

Abstract

We extend convergence theorems of discrete parameter operator-valued martingale to those of operatorvalued martingale indexed by directed set. We prove that: (1) $L^1$-bounded weak operator-valued martingale indexed by directed set converges in probability to a weak random operator. If it is uniformly integrable on a dense subset of a Hilbert space, then it is closable and converges in $L^1$ mean; (2) If a strong operator-valued martingale indexd by directed set is $L^1$ bounded and its inverse image of a compact positive symmetric operator is bounded in probability, then it converges in probability to a strong random operator. Furthermore, if it is $L^2$-bounded, then it converges to a strong random operator in $L^2$ mean; (3) An operator-valued martingale indexed by directed set which is uniformly integrable and satisfies additional condition converges in probability to a bounded random operator.