A weak equivalence theorem for weak random elements with values in weakly compactly generated Banach spaces and its applications (Q1913912)

The author proves the following Theorem. Given a mapping \(V\) from a probability space \(\Omega\) into a weakly compactly generated Banach space \(B\), which is weakly measurable (but not necessarily bounded), then there exists \(\widetilde V:\Omega\to B\) such that i) \(\widetilde V\) is a.e. equal to the norm-limit of a sequence of measurable maps \(V_n:\Omega\to B\) with finite range \((n\in\mathbb{N})\), ii) For each \(f\in B^*\), \(f\circ V=f\circ\widetilde V\) a.e. \(\widetilde V\) is a.e. uniquely determined. An application in terms of reproducing kernel Hilbert spaces is given.