On Weak Compactness in Spaces of Measures

✍ Scribed by Xiao-Dong Zhang

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 782 KB
• Volume: 143
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.2983

✦ Synopsis

It is proved that a weak* compact subset A of scalar measures on a _-algebra is weakly compact if and only if there exists a nonnegative scalar measure * such that each measure in A is *-continuous (such a measure * 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 * such that each measure in A is *-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.

1997 Academic Press Hahn Saks theorem concludes that weak* compactness implies uniform countably additivity, which in turn implies the weak compactness of the sequence (see [1, Thm 13, p. 92]). This observation leads us to investigate under which condition a weak* compact subset of measures is weakly article no. FU962983