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Ideal exhaustiveness, weak convergence and weak compactness in Banach spaces. (English) Zbl 1276.28024

Let \(\mathcal{I}\) be an ideal on \(\mathbb{N}\). Some types of \(\mathcal{I}\)-compactness are defined. Relations between \(\mathcal{I}\)-exhaustiveness and equicontinuity of measures are investigated. Some versions of limit theorems involving \(\mathcal{I}\)-pointwise convergence of measure sequences are established. An application to study the convergence in the space \(L^{\infty}(\lambda)\), for \(\lambda\) being a regular measure, is presented.

MSC:

28B15 Set functions, measures and integrals with values in ordered spaces
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
40A35 Ideal and statistical convergence
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References:

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