Jensen’s Inequality for Conditional Expectations in Banach Spaces

Abstract

In this note we present a simple proof of the inequality $\Phi \left( E^{\mathcal{A}}\xi \right) \leq E^{\mathcal{A}}\Phi (\xi )$ a.s. for separable random elements $\xi \mathcal{I}n L_{1}(\Omega ,\mathcal{F},P;X)$ in a Banach space $X,$ where $E^{\mathcal{A}}\left(\cdot\right) $ denotes conditional expectation with respect to the $\sigma $-field $\mathcal{A} \subset \mathcal{F}$, and $\Phi :X\rightarrow \mathbb{R}$ is a convex functional satisfying certain additional assumptions which are less restrictive than known till now. Some consequences of the above result are also discussed; e.g., it is shown that if $\xi $ is a Gaussian random element in $X$, then there exists a constant $0<c< \infty $ such that for each $\sigma $-field $\mathcal{A}_{0}\subset \mathcal{F}$ the family $\left\{ \exp \{c\left\| E^{\mathcal{A}}\xi \right\| ^{2}\}\mathcal{A}_{0}\subseteq \mathcal{A} \subseteq \mathcal{F}\right\} $ is uniformly integrable.