On conditional expectations in L^p(mu;L^q(nu;X))

Abstract

Let $(A,\mathcal{A},\mu )$ and $(B,\mathcal{B},\nu )$ be probability spaces, let $\mathcal{F}$ be a sub-$\sigma$-algebra of the product $\sigma$-algebra $\mathcal{A}\times \mathcal{B}$, let $X$ be a Banach space, and let $1<p,q<\infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\mathcal{F}$ defines a bounded linear operator from $L^p(\mu ;L^q(\nu ;X))$ onto $L_{\mathcal{F}}^p(\mu ;L^q(\nu ;X))$, the closed subspace in $L^p(\mu ;L^q(\nu ;X))$ of all functions having a strongly $\mathcal{F}$-measurable representative.

A further corollary has been added