Zhang, Xiao-Dong On weak compactness in spaces of measures. (English) Zbl 0874.46030 J. Funct. Anal. 143, No. 1, 1-9 (1997). It is proved that a weak\(^*\) compact subset \(A\) of scalar measures on a \(\sigma\)-algebra is weakly compact if and only if there exists a nonnegative scalar measure \(\lambda\) such that each measure in \(A\) is \(\lambda\)-continuous (such a measure \(\lambda\) is called a control measure for \(A\)). This result is then used to obtain a very general form of the Vitali-Hahn-Saks theorem on finitely additive vector measures. Finally, it is proved that a weak\(^*\) compact subset \(A\) of regular Borel measures on an \(F\)-space is weakly compact if and only if there exists a nonnegative regular Borel measure \(\lambda\) such that each measure in \(A\) is \(\lambda\)-continuous. This latter result shows that Grothendieck’s theorem on weak\(^*\) convergent sequences of measures is valid not only for weak\(^*\) convergent sequences but also for weak\(^*\) compact subsets with a control measure. Reviewer: X.-D.Zhang (Boca Raton) Cited in 14 Documents MSC: 46G10 Vector-valued measures and integration 46E27 Spaces of measures 28B05 Vector-valued set functions, measures and integrals 46A50 Compactness in topological linear spaces; angelic spaces, etc. Keywords:weak compactness; subset of scalar measures on a \(\sigma\)-algebra; control measure; Vitali-Hahn-Saks theorem on finitely additive vector measures; regular Borel measure; Grothendieck’s theorem on weak\(^*\) convergent sequences of measures PDF BibTeX XML Cite \textit{X.-D. Zhang}, J. Funct. Anal. 143, No. 1, 1--9 (1997; Zbl 0874.46030) Full Text: DOI Link OpenURL