Two Embedding Theorems with Applications to Weak Convergence and Compactness in Spaces of Additive Type Functions

Porcelli, Pasquale

doi:10.1512/iumj.1960.9.59016

Cited by 13 publications

(12 citation statements)

References 7 publications

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“…Let X denote this limit. By using the unconditional convergence property of ^-bounded measures [8, 2.3], the Orlicz-Pettis theorem, and a result of Procelli [7], we can show that the sequence A,-converges weakly to X in ba(Su Soi). By a result of Leader [5] this implies that the X; are uniformly absolutely continuous with respect to a positive finitely additive bounded set function <p defined on S 0 i.…”

mentioning

confidence: 99%

On the existence of a control measure for strongly bounded vector measures

Brooks¹

1971

Bull. Amer. Math. Soc.

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mentioning

confidence: 99%

On the existence of a control measure for strongly bounded vector measures

Brooks¹

1971

Bull. Amer. Math. Soc.

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“…[6]) on weak convergence and sequential weak compactness in H(X, T) to obtain results such as the following (X,S,g) and ||L|| = ||fe||, where the limit is taken (cf. [5]) in the Moore-Smith sense over the directed set of partitions π of X in T. But, and this is the crux of the matter, if fe^* then F(r) = F(l)-r for each re &.…”

Section: R B Darst That C*(x T) Is Isomorphic and Isometric To H(xmentioning

confidence: 99%

A continuity property for vector valued measurable functions

Darst¹

1962

Pacific J. Math.

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“…Porcelli's original proof [15] was extremely long and gruelling, while the direct proof due to Darst [5] still contains many technical arguments. The present proof is less elementary, but is devoid of technicalities and directly relates the finitely additive result to its analogue for countably additive measures.…”

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confidence: 99%

The Carathéodory extension theorem for vector valued measures

Kupka¹

1978

Proc. Amer. Math. Soc.

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Abstract. This paper comprises three advertisements for a known theorem which, the author believes, deserves the title of the Caratheodory extension theorem for vector valued premeasures. Principal among these is a short and transparent proof of Porcelli's criterion for the weak convergence of a sequence in the Banach space of bounded finitely additive complex measures defined on an arbitrary field, and equipped with the total variation norm. Also, a characterization of the so-called Caratheodory Extension Property is presented, and there is a brief discussion of the relevance of this material to stochastic integration.Amongst the numerous criteria for the extendability of a vector valued premeasure (see [9] for an admirable summary and comprehensive bibliography), precisely one stands out both for its simplicity and for its applicability. Our aim in this paper is to rescue this criterion from its relative obscurity within the speciahst literature by providing three "advertisements" for it. The main advertisement will be a short, transparent proof of Porcelli's characterization of the weak convergence of a sequence of finitely additive complex measures (Theorem 2). The second advertisement will be a simple characterization of those Banach spaces for which the classical Caratheodory extension theorem remains essentially valid (Theorem 5), and the third advertisement will be an indication of the usefulness of the criterion to the construction of stochastic integrals.We begin with a brief discussion of the criterion itself. Recall that a (Banach space valued) set function u is said to be strongly bounded if we have lim"^w p(E") = 0 for every infinite sequence {£"} of pairwise disjoint sets in the domain of ¡i, and that ju is a premeasure if we have li(E)=^p(E") 71 for every finite or infinite sequence {E"} of pairwise disjoint sets in the domain of u whose union, E, also lies in the domain of u. Here now is the theorem which, we feel, deserves the title of the Caratheodory extension theorem for Banach space valued premeasures.

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Two Embedding Theorems with Applications to Weak Convergence and Compactness in Spaces of Additive Type Functions

Cited by 13 publications

References 7 publications

On the existence of a control measure for strongly bounded vector measures

On the existence of a control measure for strongly bounded vector measures

A continuity property for vector valued measurable functions

The Carathéodory extension theorem for vector valued measures

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