On Birkhoff integrability for scalar functions and vector measures
type: Item
instance of: scholarly article
title: On Birkhoff integrability for scalar functions and vector measures
zbMATH Open document ID: 1181.28014
published in: Monatshefte für Mathematik
publication date: 2009-06-30T00:00:00Z
review text: \textit{R. G. Bartle, N. Dunford} and \textit{J. Schwartz} [Can. J. Math. 7, 289--305 (1955; Zbl 0068.09301)] developed an integral for scalar functions with respect to vector measures as an analogue of the Riesz Representation Theorem for compact operators on Banach spaces of continuous functions on a compact Hausdorff space. In this paper this integral is related to the S*-integral studied intensively by \textit{I. Dobrakov} [Czech. Math. J. 38(113), No. 3, 434--449 (1988; Zbl 0674.28003)]. It is shown that a scalar function \(f\) is B-D-S integrable with respect to a vector measure \(m\) iff it is S*-integrable, which in turn is also equivalent to saying that \(f\) is measurable and there is a partition of the domain into countably many measurable sets such that, for every finer, countable, partition \(\{A_n\}_n\) and any choice of points \(\omega_n \in A_n\), the series \(\sum_n f(\omega_n)m\{A_n\}\) is unconditionally convergent.
reviewed by: Srinivasa Swaminathan
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 5571614
zbMATH Keywords: vector measures
zbMATH Keywords: Bartle-Dunford-Schwartz integral
zbMATH Keywords: Birkhoff integral
author: Antonio Fernández
author: Fernando Mayoral
author: Francisco Naranjo
author: J. Rodríguez
MaRDI profile type: Publication
full work available at URL: s00605-008-0020-z
OpenAlex ID: W2083123129
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DOI: 10.1007/S00605-008-0020-Z
description: scientific article; zbMATH DE number 5571614