1 Introduction and preliminaries The aim of this note is to prove the following basic theorem: Let (Ω, σ, u) be a probability space, (B, ‖·‖) a weakly compactly generated Banach space, and a mapping V from Ω...1 Introduction and preliminaries The aim of this note is to prove the following basic theorem: Let (Ω, σ, u) be a probability space, (B, ‖·‖) a weakly compactly generated Banach space, and a mapping V from Ω to B be a weak random element, then there exists a unique strongly measurable random element V from Ω to B under the sense of almost sure equality such that (?) is weakly equivalent to the weak random dement V. This theorem itself not only removes the limitation that the weak random element considered in a theorem due to Lewis is bounded, but also has many applications to probability theory in Banach spacest. As an example of applications, we give a theorem of properties of the reproducing kernel Hilbert spaces for weak twofold weak random elements.展开更多
文摘1 Introduction and preliminaries The aim of this note is to prove the following basic theorem: Let (Ω, σ, u) be a probability space, (B, ‖·‖) a weakly compactly generated Banach space, and a mapping V from Ω to B be a weak random element, then there exists a unique strongly measurable random element V from Ω to B under the sense of almost sure equality such that (?) is weakly equivalent to the weak random dement V. This theorem itself not only removes the limitation that the weak random element considered in a theorem due to Lewis is bounded, but also has many applications to probability theory in Banach spacest. As an example of applications, we give a theorem of properties of the reproducing kernel Hilbert spaces for weak twofold weak random elements.
