On the Radon-Nikodym theorem
type: Item
instance of: scholarly article
title: On the Radon-Nikodym theorem
zbMATH Open document ID: 0861.28009
DOI: 10.1007/BF00052195
author: Z. Magyar
published in: Acta Mathematica Hungarica
publication date: 1997-04-21T00:00:00Z
review text: Radon-Nikodým/Lebesgue decomposition theorems are investigated from the perspective of upper integrals. Given an outer integral \(N\) on a set \(X\) and a vector sublattice \(V\) of the lattice of nonnegative functions on \(X\) such that \(N\) is additive and finite on \(V^+\) there is a natural \(L_1\) space and another upper integral \(\overline {N}\) given by \(\overline {N}(f)= \inf N(g)\), taken over \(L_1\) functions \(g\geq f\). The essentialization of an upper integral \(N\) is \(\widehat {N}(f)= \sup N(f\cdot 1_A))\), taken over measurable sets \(A\) of finite measure. For two Bourbaki integrals \(I\) and \(J\) defined on a Stonean vector sublattice of the lattice of nonnegative functions on \(X\), let \(\widehat{I}\) and \(\widehat {J}\) be the corresponding essentializations of the upper integrals determined by \(I\) and \(J\). Then there is a measurable set \(A\), unique up to negligible sets, and a nonnegative measurable function \(h\), unique almost everywhere-\(I\), such that \(X\setminus A\) is \(\widehat{I}\) negligible and \(\widehat{J} (fh)=\widehat{I} (f\cdot 1_A)\) for all \([0,\infty]\)-valued functions \(f\) on \(X\). The result also holds for Daniell integrals when restricted to sigma finite essentializations, but is not true in general. These results were previously known when \(X\) is a locally compact Hausdorff space and \(V\) is the vector lattice of compactly supported continuous functions. Another Radon-Nikodým/Lebesgue decomposition theorem is given for more general upper integrals, assuming a type of decomposability.
Mathematics Subject Classification ID: 28C05
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28A50
zbMATH DE Number: 897173
zbMATH Keywords: Radon-Nikodým theorem
zbMATH Keywords: Lebesgue decomposition theorems
zbMATH Keywords: Bourbaki integrals
zbMATH Keywords: Daniell integrals
zbMATH Keywords: upper integrals
reviewed by: J. W. Hagood
MaRDI profile type: Publication
cites work: Q4770999
cites work: Q5570501
cites work: Q4338930
cites work: Q5768777
cites work: Equivalences of Measure Spaces
full work available at URL: https://doi.org/10.1007/bf00052195
OpenAlex ID: W1980169041
description: scientific article; zbMATH DE number 897173