Lifting Approach to Integral Representations of Rich Multimeasures with Values in Banach Spaces II

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.