On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation

AbstractBanach space valued multifunctions defined on a complete σ-finite measure space (Ω, Σ, μ) are studied. Their set valued integral is defined and its properties are examined. Since the definition of the integral involves the set of integrable selectors of the multifunction, the structure of that set is also studied. Some Banach-like spaces of multifunctions are introduced and studied. Multifunctions depending on a parameter are also considered and it is examined wheter certain continuity, semicontinuity and other topological properties are preserved by set valued integration. Finally, for integrable multifunctions, the properties of their set valued conditional expectation are studied