On functions that are universally Pettis integrable
This paper is concerned with a class of functions defined on a compact Hausdorff space K and taking values in the dual of a Banach space E. One of the main results is that if a bounded function \(f: K\to (E^*,weak^*)\) is universally Lusin measurable, then it is universally Pettis integrable whenever it is universally scalarly measurable. Several applications are given. For example, if C is a weak*-compact subset of the dual \(E^*\) such that the identity map \(I: (C,weak^*)\to E^*\) is universally scalarly measurable, then the same is true for the identity map \(I: (w^*-cl(conv(C)),weak^*)\to E^*\). If the linear span of such a set C is norm dense in \(E^*\), then \(E^*\) has the weak Radon-Nikodym property. If \(f: K\to (E^*,weak^*)\) is universally Lusin measurable, then f is universally Pettis integrable if and only if for every Radon probability measure \(\lambda\) on K, the set \(\{<f,x>:x\in E,\| x\| \leq 1\}\) is almost weakly precompact in \(L_{\infty}(K,\lambda).\) The second half of the paper discusses the relations between Pettis integrability of a function and the Bourgain property. The Bourgain property always implies that the function is Pettis integrable, but the converse is not always true. However, if K is a compact metric space and E is a separable Banach space, then \(f: K\to E^*\) is universally Pettis integrable if and only if f has the Bourgain property for every Radon probability measure on K.
functions defined on a compact Hausdorff space
taking values in the dual of a Banach space
scientific article; zbMATH DE number 3921274
scientific article