Continuity of conditional expectation in Orlicz spaces

TL;DR

This paper investigates the conditions under which conditional expectation operators are continuous in Orlicz spaces, extending known results from classical $L^p$-spaces to more general settings.

Contribution

It provides necessary and sufficient conditions for the $L^{oldsymbol{ extphi}}$-convergence of conditional expectations in Orlicz spaces, generalizing previous $L^p$-space results.

Findings

Established conditions for convergence of conditional expectations in Orlicz spaces.

Generalized classical $L^p$ results to Orlicz space setting.

Extended understanding of conditional expectation continuity in more general function spaces.

Abstract

The continuity of conditional expectation on Orlicz spaces is investigated. Indeed, we provide some necessary and sufficient conditions on a sequence $\{\mathcal{A}_n\}_{n\in\mathbb{N}}$ of $\sigma$-subalgebras for $L^{\varphi}$-convergence of the related conditional expectations. Our results generalize similar results in $L^p$-spaces.