Order Convergence of Martingales in Terms of Countably Additive and Purely Finitely Additive Martingales

Abstract

Let $(E, \mathscr{B}, \mu)$ be a measure space, let $\theta$ be a directed set with a countable cofinal subset, and let $(\mathscr{B}_\tau)_{\tau \in \theta}$ be an increasing family of sub-$\sigma$-algebras of $\mathscr{B}$. A martingale $(f_\tau)_{\tau \in \theta}$ is said to be of semibounded variation whenever the set $\{\int_B f_\tau d\mu \mid \tau \in \theta, B \in \mathscr{B}_\tau\}$ is bounded either from above or below. Denote conditional expectation by $\mathscr{E}$. We show that if every martingale of the form $\mathscr{E}(f \mid \mathscr{B}_\tau)_{\tau\in\theta}$ for some $\mathscr{B}$-measurable function $f$ with $\int|f| d\mu < \infty$ is order convergent, then every martingale of semibounded variation is order convergent. When the family $(\mathscr{B}_\tau)_{\tau\in\theta}$ satisfies a certain refinement condition, we obtain a sufficient condition for order convergence of martingales of semibounded variation in terms of order convergence of martingales which converge stochastically to 0.