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On weak compactness in spaces of measures. (English) Zbl 0874.46030

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.

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.
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