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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