On Weak Compactness in Spaces of Measures

It is proved that a weak* compact subsetAof scalar measures on aσ-algebra is weakly compact if and only if there exists a nonnegative scalar measureλsuch that each measure inAisλ-continuous (such a measureλis called a control measure forA). 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 subsetAof regular Borel measures on anF-space is weakly compact if and only if there exists a nonnegative regular Borel measureλsuch that each measure inAisλ-continuous. This latter result shows that Grothendieckʹs 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.