Radon-Nikodým property of conjugate Banach spaces and \(w^*\)-equivalence theorems of \(w^*-\mu\)-measurable functions. (English) Zbl 0868.46014

Summary: A deep representation theorem of random conjugate spaces and 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 \(B\), \((\Omega,\sigma,\mu)\) be a given probability space. Then every \(B^*\)-valued \(w^*\)-\(\mu\)-measurable function defined on \((\Omega,\sigma,\mu)\) is \(w^*\)-equivalent to a \(B^*\)-valued \(\mu\)-measurable function defined on \((\Omega,\sigma,\mu)\) if and only if \(B^*\) has the Radon-Nikodým property with respect to \((\Omega,\sigma,\mu)\).


46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46G10 Vector-valued measures and integration