Norming sets and integration with respect to vector measures
Let \(\nu\) be a countably additive measure defined on a measurable space \((\Omega,\Sigma)\) and taking values in a Banach space \(X\). Then, for each functional \(x^*\in X^*\), \(\nu_{x^*}(A):=\langle \nu(A),x^*\rangle,\,A\in \Sigma, \) is a countably additive real measure. A measurable function \(f:\Omega\to \mathbb{R}\) is called \textit{weakly integrable} (with respect to \(\nu\)) if \(f\in \mathcal{L}^1(\nu_{x^*})\) for each \(x^*\in X^*\). A weakly integrable function is \textit{integrable} provided for each \(A\in \Sigma\) there is an element of \(X\) (denoted by \(\int_A f\,d\nu\)) such that \(\langle\int_A f\,d\nu,x^*\rangle = \int_Af\,d\nu_{x^*}\) for every \(x^*\in X^*\). The authors investigate the problem: When it is enough to test the integrability (weak integrability) of \(f\) on a norming subset \(\Lambda\) instead of all \(X^*\)? In particular, they show that this is the case if \(\Lambda\) is a James boundary for the unit ball \(B_{X^*}\) (resp. \(\Lambda\) is w*-thick). A set \(\Lambda \subset X^*\) is \textit{w*-thick} if it is not w*-thin, and is w*-thin if we can write \(\Lambda=\bigcup_1^{\infty}\Lambda_n\), where \(\Lambda_1\subset\Lambda_2\dots\), and for every \(n\) \[ \inf_{\|x\|=1}\sup_{x^*\in\Lambda_n}|\langle x,x^*\rangle|=0. \] Examples, showing that these results are not valid for all norming sets, are presented.
scientific article; zbMATH DE number 5526740
scientific article