Short Proofs of Two Convergence Theorems for Conditional Expectations

Abstract

In this paper there are given new proofs of two convergence theorems for conditional expectations, concerning convergence in measure and convergence almost everywhere of a sequence of conditional expectations $P_n^\mathscr{F}0f$ of a bounded function $f$, given a $\sigma$-field $\mathscr{F}_0$, with respect to varying probability measures $P_n$.