Journal of Convex Analysis 31 (2024), No. 3, 867--888
Copyright Heldermann Verlag 2024
Lifting Approach to Integral Representations of Rich Multimeasures with Values in Banach Spaces II
Kazimierz Musial
Institute of Mathematics, Wroclaw University, Wroclaw, Poland
kazimierz.musial@math.uni.wroc.pl
If X is a separable Banach space with RNP and Γ is an Effros measurable and Pettis integrable multifunction, then the conditional expectation of Γ with respect to a sub-σ-algebra is well described by the set of conditional expectations of selections of Γ. This is possible due to the Castaing representation of Γ that fails in case of non-separable X. Instead of RNP and separability of X we assume that the multimeasure defined by the Pettis integral of Γ is rich in selections possessing strongly measurable Radon-Nikodym derivatives. In general that cannot be reduced to a separable space. Then, using a lifting, we prove the existence of an Effros measurable conditional expectation of Γ in case of an arbitrary non-separable X and present its representation in terms of quasi-selections of Γ. We apply then the description to martingales of Pettis integrable multifunctions obtaining a scalarly equivalent martingale of measurable multifunctions with many martingale selections.
Keywords: Multimeasures, multifunctions, Pettis integral, Radon-Nikodym property, conditional expectation, martingales, rich multimeasures.
MSC: 28B20; 28B05, 46G10, 46B22.
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