Families of vector measures of uniformly bounded variation (Q869253)

Let \(X\) be a Banach space, \(\mathcal F \subset \mathcal P (\Omega)\) an algebra over a non-empty set \(\Omega\), and \(\{F_{\tau}\} = \{F_{\tau} : \tau \in T\}\) a family of \(X\)-valued vector measures defined on \(\mathcal F\). A simplified proof is given of the Nikodým boundedness theorem [\textit{J.\,Diestel} and \textit{J.\,J.\thinspace Uhl}, ``Vector measures'' (Math.\ Surv.\ 15, AMS, Providence/RI) (1977; Zbl 0369.46039)], which says that if \(\mathcal F\) is a \(\sigma\)-algebra and each \(\{F_{\tau}\}\) be setwise bounded, then the family \(\{F_{\tau}\}\) is uniformly bounded.