Almost everywhere continuity of conditional expectations

TL;DR

This paper establishes a necessary and sufficient condition on sequences of sigma-subalgebras that guarantees almost everywhere convergence of their associated conditional expectations.

Contribution

It provides a precise criterion for the almost everywhere convergence of conditional expectations based on sigma-subalgebra sequences.

Findings

Characterizes when conditional expectations converge almost everywhere.

Provides a necessary and sufficient condition for convergence.

Enhances understanding of the behavior of conditional expectations in probability theory.

Abstract

A necessary and sufficient condition on a sequence $\{\mathfrak{A}_n\}_{n\in \mathbb{N}}$ of $\sigma$-subalgebras that assures convergence almost every where of conditional expectations is given.