Radon-Nikodym property of conjugate Banach spaces and w~*-equivalence theorems of w~*-μ-measurable functions

摘要

A deep representation theorem of random conjugate spaces and its several important applications are given. As an application of the representation theorem, the following basic theorem is also proved: let B~* be the conjugate space of a Banach space B, (Ω, σ, μ) be a given probability space. Then every B~*-valued w~*-μ-measurable function defined on (Ω, σ, μ) is w~*-equivalent to a B~*-valued μ-measurable function defined on (Ω, σ, μ) if and only if B~* has the Radon-Nikodym property with respect to (Ω, σ, μ).