  Let $(A,\mathscr{A},\mu)$ and $(B,\mathscr{B},\nu)$ be probability spaces,
let $\mathscr{F}$ be a sub-$\sigma$-algebra of the product $\sigma$-algebra
$\mathscr{A}\times\mathscr{B}$, 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 $\mathscr{F}$ defines a bounded linear
operator from $L^p(\mu;L^q(\nu;X))$ onto $L^p_{\mathscr{F}}(\mu;L^q(\nu;X))$,
the closed subspace in $L^p(\mu;L^q(\nu;X))$ of all functions having a strongly
$\mathscr{F}$-measurable representative.
