On weak compactness in spaces of measures (Q677462)

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.