On functions that are universally Pettis integrable
type: Item
instance of: scholarly article
title: On functions that are universally Pettis integrable
zbMATH Open document ID: 0576.46034
author: Lawrence H. Riddle
author: Elias Saab
published in: Illinois Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B22
zbMATH DE Number: 3921274
zbMATH Keywords: functions defined on a compact Hausdorff space
zbMATH Keywords: taking values in the dual of a Banach space
zbMATH Keywords: universally Lusin measurable
zbMATH Keywords: universally Pettis integrable
zbMATH Keywords: universally scalarly measurable
zbMATH Keywords: weak Radon-Nikodym property
zbMATH Keywords: Radon probability measure
zbMATH Keywords: almost weakly precompact
zbMATH Keywords: Pettis integrability
zbMATH Keywords: Bourgain property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3921274