Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.