English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper an extension of the Brook's control measure existence theorem for families of vector measures is established. This result is applied to pointwise convergent sequences of finitely additive vector measures to obtain a generalization of the Vitali-Hahn-Saks theorem. Introduction. The results presented in this paper began with an attempt to extend the classical Vitali-Hahn-Saks theorem to sequences of finitely additive vector measures. The work of Bogdanowicz [8], Drewnowski [5], and Labuda [6] indicated that the Baire category argument could be used to establish a vector form of the Vitali-Hahn-Saks theorem for sequences convergent on a delta ring (as opposed to a a-algebra) provided the delta ring was endowed with an appropriate sequentially complete topology. Earlier, Ando [1], using a lemma due to Phillips [9] as a substitute for the Baire category argument, noted the validity of the Vitali-Hahn-Saks theorem for sequences of finitely additive scalar measures on a a-algebra. Later, Brooks and Jewett [4] generalized Phillips' lemma to vector functions and obtained a Vitali-Hahn-Saks theorem for sequences of strongly bounded vector measures convergent pointwise on a a-algebra. Combining the methodology developed by Ando with the Brooks-Jewett formulation of Phillips' lemma yields a formulation of the Vitali-Hahn-Saks theorem which pulls together the ideas developed in the above listed extensions. The results also lead to a more refined knowledge of the control measure existence theorems developed by Bartle, Dunford, and Schwartz [2], Brooks [3], and Drewnowski [5]. Let V denote a ring of subsets of an abstract space X, let N denote the positive integers and let (R +) R denote the (nonnegative) reals. Denote by C(V) the space of all subadditive and increasing functions, from the ring V into R +, which are zero at the empty set. The space C ( V) is called the space of contents on the ring V and elements are referred to as contents. A sequence of sets An E V, n E N, is said to be dominated if there exists a set B E V such that An C B for n = 1, 2, 3, . A contentp E C( V) is said to be Rickart on the ring V if limnp(An) = 0 for each dominated, disjoint sequence An e V, n E N. A set of contents P c C (V) is said to be uniformly Rickart on the ring V if the above limit holds uniformly with Received by the editors August 11, 1976. AMS (MOS) subject classifications (1970). Primary 28A45, 28A10, 28A60. ? American Mathematical Society 1978

Generalizations of the Vitali-Hahn-Saks theorem on vector measures