On Finitely Additive Vector Measures

Brooks, James K.; Jewett, Robert S.

doi:10.1073/pnas.67.3.1294

Cited by 60 publications

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“…The following proposition has been established by Brooks and Jewett [4] for sequences of finitely additive, strongly bounded vector measures on a a-algebra. The Brooks-Jewett arguments carry over to sequences of Rickart vector charges converging pointwise on a delta ring.…”

Section: Corollarymentioning

confidence: 99%

“…The fundamental result is the Banach space formulation of Phillips' lemma [10], which Brooks and Jewett [4] have established.…”

Section: Corollarymentioning

confidence: 99%

“…The vector formulation of the Vitali-Hahn-Saks theorem requires a discussion of the properties of pointwise convergent sequences of Rickart vector charges as developed by Brooks and Jewett [4]. The fundamental result is the Banach space formulation of Phillips' lemma [10], which Brooks and Jewett [4] have established.…”

Section: Corollarymentioning

confidence: 99%

“…The vector form of Phillips' lemma [10] established by Brooks and Jewett [4] is used to show that each sequence of finitely additive, Rickart vector charges, converging pointwise on a delta ring of sets, is uniformly Rickart. This result is combined with the control charge existence theorems to establish a Vitali-Hahn-Saks theorem which contains the Bogdanowicz version [8], obtained independently by Labuda [6], and the Brooks-Jewett extension [4].…”

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confidence: 99%

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Generalizations of the Vitali-Hahn-Saks Theorem on Vector Measures

Oberle¹

1977

Proceedings of the American Mathematical Society

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Abstract.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 jV 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

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Section: Corollarymentioning

confidence: 99%

“…The fundamental result is the Banach space formulation of Phillips' lemma [10], which Brooks and Jewett [4] have established.…”

Section: Corollarymentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Generalizations of the Vitali-Hahn-Saks Theorem on Vector Measures

Oberle¹

1977

Proceedings of the American Mathematical Society

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“…As we indicated in Theorem 5.1, this example answers two other questions dealing with vector measures. In [9], using Rickart's notion of sboundedness, Brooks and Jewett showed that the pointwise limit of a sequence of s-bounded set functions was s-bounded. This result fails for representing measures.…”

Section: Y-imentioning

confidence: 99%