Almost sure equiconvergence of conditional expectations (Q762288)

If (X,\({\mathcal F},P)\) is a probability space then a pseudo-metric \(\delta\) can be defined on the sub-\(\sigma\)-fields of \({\mathcal F}\) by \[ \delta ({\mathcal A},{\mathcal B})=\sup_{A\in {\mathcal A}}\inf_{B\in {\mathcal B}}P(A\Delta B)\bigvee \sup_{b\in {\mathcal B}}\inf_{A\in {\mathcal A}}P(A\Delta B). \] Boylan, Neveu, and Rogge, among others, have considered equiconvergence of conditional expectations of uniformly bounded measurable functions given sub-\(\sigma\)-fields \(\{\) \({\mathcal F}_ n:1\leq n\leq \infty \}\) in probability and in \(L_ p\), \(1\leq p<\infty\), as \(\delta\) (\({\mathcal F}_ n,{\mathcal F}_{\infty})\to 0\). This paper proves the corresponding almost sure equiconvergence results when \({\mathcal F}_ n\uparrow {\mathcal F}_{\infty}\) or \({\mathcal F}_ n\downarrow {\mathcal F}_{\infty}.\) A sharp uniform bound for the rate of convergence is given. A consequence is that if \({\mathcal F}_ n\uparrow {\mathcal F}_{\infty}\) or \({\mathcal F}_ n\downarrow {\mathcal F}_{\infty}\) then the sequence of conditional expectations given \({\mathcal F}_ n\) converges uniformly for all uniformly bounded measurable functions to the conditional expectation given \({\mathcal F}_{\infty}\) if and only if \(\delta ({\mathcal F}_ n,{\mathcal F}_{\infty})\to 0.\)