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

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Abstract

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 Lü, 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.).

Original languageEnglish
Pages (from-to)1-8
Number of pages8
JournalPositivity: an international journal devoted to the theory and applications of positivity in analysis
DOIs
Publication statusE-pub ahead of print - 15 Feb 2017

Keywords

  • Conditional expectations in $$L^p(\mu ;L^q(\nu ;X))$$Lp(μ;Lq(ν;X))
  • Dual of $$L_\mathscr {F}^p(\mu ;L^q(\nu ;X))$$LFp(μ;Lq(ν;X))
  • Radon–Nikodým property

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