On conditional expectations in L p (μ; L q (ν; X))
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Abstract
Let (A, A, μ) and (B, B, ν) be probability spaces, let F be a sub-σ-algebra of the product σ-algebra A× B, let X be a Banach space and let 1 < p, q< ∞. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to F defines a bounded linear operator from L p (μ; L q (ν; X)) onto LFp(μ;Lq(ν;X)), the closed subspace in L p (μ; L q (ν; X)) of all functions having a strongly F-measurable representative.
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