Conditional expectations in Lp(μ;Lq(ν;X))

Let (Formula presented.) and (Formula presented.) be probability spaces and X a Banach space. We prove that for all (Formula presented.), the conditional expectation with respect to any sub-(Formula presented.)-algebra (Formula presented.) of the product (Formula presented.)-algebra (Formula presented.) defines a bounded linear operator from (Formula presented.) onto (Formula presented.), the closed subspace in (Formula presented.) of all functions having a strongly (Formula presented.)-measurable representative. As an application we obtain a simple proof of the following result of Lu, Yong, and Zhang: if (Formula presented.) has the Radon–Nikodým property, then for all (Formula presented.) we have (Formula presented.) with equivalent norms ((Formula presented.)). These results are shown to be optimal in the following sense: (i) the conditional expectation need not be contractive; (ii) the duality does not extend to the pair (Formula presented.).