The Vitali-Hahn-Saks theorem for algebras

Abstract

The Vitali-Hahn-Saks and Nikodym Theorems are some of the most important and useful theorems in measure theory [8, 111.7.2, 111.7.4, IV.9.8; 5, 1.3.1, 1.48, 1.4.101. There have been many generalizations of these results in various directions; for example, the values of the measures are assumed to be in a topological vector space or topological group [S, 4, 31 or the domains of the measures are assumed to be special types of algebras which are not necessarily a-algebras [lo]. In [lo], Schachermayer has made a detailed study of the Vitali-Hahn-Saks and Nikodym Boundedness theorems for scalar-valued measures defined on algebras (even Boolean algebras). In particular, Schachermayer has shown that the Vitali-HahnSaks (VHS) property for algebras implies the Nikodym Boundedness (NB) property for algebras [ 10, 2.51 and, moreover, has given a suflicient condition (Property (E) of [ 10, 4.21) for an algebra to satisfy the VHS property. Haydon [9] has considered a property called the Sequential Completeness Property (SCP) which is more general than Schachermayer’s property (E), has shown that the Grothendieck (G) property is valid for algebras satisfying SCP, and has stated that the VHS property holds for algebras which satisfy SCP [9, lB]. The methods of Haydon are nontrivial, using Rosenthal’s Lemma and Stone space methods, and yield only the Grothendieck property although Haydon states that the VHS property for algebras satisfying SCP can be obtained by using more care in the arguments. In this note, we show that the matrix methods employed in [ 1 ] (see also [2]) can be utilized to show that algebras of sets which satisfy SCP also satisfy the VHS property. It should be noted that these matrix methods are of a significantly simpler nature than the methods employed 116 0022-247X/85 $3.00