Birkhoff integral for multi-valued functions
type: Item
instance of: scholarly article
title: Birkhoff integral for multi-valued functions
zbMATH Open document ID: 1066.46037
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: In this paper, \(X, X^{*}\) denotes a Banach space and its topological dual and \(B_{X}, B_{X^{*}} \) the closed unit balls of \(X, X^{*}\); \( (\Omega , \Sigma, \mu)\) is a complete finite measure space, \(cwk(X)\) is the family of all non-empty, weakly compact, convex subsets of \(X\) with Hausdorff distance \(h\), \(ck(X)\) is the family of all non-empty, compact, convex subsets of \(X\) with Hausdorff distance \(h\), and \( \ell_{\infty}(B_{X^{*}}) \) the Banach space of all bounded real-valued functions on \(B_{X^{*}}\). It is well-known that there is a mapping \( j: cwk(X) \to \ell_{\infty}(B_{X^{*}}) \), with \[ j(A)(x^{*})= \sup \{ x^{*} (x): x \in A \} , \] which is additive, positively homogeneous, distance preserving and with \( j(cwk(X))\) closed in \(\ell_{\infty}(B_{X^{*}}) \). For a Banach space \(Y\), an \( f: \Omega \to Y \) is said to be summable with respect to a countable partition \(\Gamma = \{ A_{n} \} \) of \(\Omega \) if \(f_{| A_{n}} \) is bounded whenever \(\mu( A_{n}) > 0 \) and the set \(J(f, \Gamma)= \{ \sum_{n} \mu ( A_{n}) f(t_{n}): t_{n} \in A_{n} \}\) consists of unconditionally convergent series. If for every \( \varepsilon >0\) there is a countable partition \( \Gamma\) of \(\Omega \) in \( \Sigma \) for which \(f\) is summable and \( \| . \| - \text{diam} (J(f, \Gamma)) \leq \varepsilon \), then the Birkhoff integral is \[ (B) \int_{\Omega} f \,d \mu = \bigcap \{ \overline{co(J(f, \Gamma))} : f \text{ summable with respect to } \Gamma \}. \] Birkhoff integrability lies between Pettis and Bochner integrability. A multi-valued \( F : \Omega \to cwk(X) \) is called Birkhoff integrable if \( j \circ F : \Omega \to \ell_{\infty}(B_{X^{*}}) \) is Birkhoff integrable. First, the authors give some equivalent characterizations of Birkhoff integrability and then prove that this definition is independent of any choice of embedding of \( cwk(X)\) into a Banach space. Then some relations between Pettis, Debreu, and Birkhoff integrability are e…
reviewed by: Surjit Singh Khurana
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28B20
zbMATH DE Number: 2107206
zbMATH Keywords: integration of multifunctions
zbMATH Keywords: Debreu integral
zbMATH Keywords: Pettis integral
zbMATH Keywords: Birkhoff integral
zbMATH Keywords: Aumann integral
author: B. Cascales
author: J. Rodríguez
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.026
OpenAlex ID: W1977661816
cites work: Fatou lemma for the Pettis integral
cites work: Integrals of set-valued functions
cites work: The largest class of closed convex valued multifunctions for which Effros measurability and scalar measurability coincide
cites work: Remarks on the set-valued integrals of Debreu and Aumann
cites work: A sequential property of set-valued maps
cites work: The Birkhoff integral and the property of Bourgain
cites work: Convex analysis and measurable multifunctions
cites work: Q5341488
cites work: Q5613035
cites work: Q3329946
cites work: Q4148368
cites work: Q4138969
cites work: On the Pettis integral of closed valued multifunctions
cites work: K‐analytic spaces with metrizable compacta
cites work: Q3598957
cites work: Q4788813
cites work: Q4433334
cites work: Integrals, conditional expectations, and martingales of multivalued functions
cites work: Measurable relations
cites work: Q4064160
cites work: Q3220809
cites work: Q5569793
cites work: Q4136031
cites work: Q4273174
cites work: On Integration in Vector Spaces
cites work: Integration in a Convex Linear Topological Space
cites work: An Embedding Theorem for Spaces of Convex Sets
cites work: On functions that are universally Pettis integrable
cites work: Espaces de Banachs faiblement K-analytiques
cites work: Q4354694
cites work: Q4512493
DOI: 10.1016/J.JMAA.2004.03.026
description: scientific article; zbMATH DE number 2107206