min and max functions, Bochner-Radon-Nikodým approximations, and a representation characterization
type: Item
instance of: scholarly article
title: min and max functions, Bochner-Radon-Nikodým approximations, and a representation characterization
zbMATH Open document ID: 0699.28002
DOI: 10.1007/BF01765935
author: William D. L. Appling
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1989-01-01T00:00:00Z
review text: Basic properties of min\(\{\) x,y\(\}\), max\(\{\) x,y\(\}\), the sum supremum and sum infimum functionals and differential equivalence are used to obtain the following for the refinement integral: (1) a form of the Bochner-Radon-Nikodým theorem (a finitely additive \(\mu\)-continuous measure can be approximated arbitrarily close in variation norm by an indefinite \(\mu\)-integral) for set-valued set-functions which does not require integrability, and (2) a characterization of real-valued functions h with domain [a,b] which admit a representation of the form \[ h(q)-h(p)=\int \max \{\min \{\alpha (I)-p,q-p\},0\}\mu (I),\quad a\leq p\leq q\leq b, \] for some nonnegative finitely additive function on a field of sets and some set-valued function \(\alpha\) on the same field. The characterization in (2) is a bounded concavity condition: \[ M\geq (h(t)-h(s))/(t-s)\geq (h(u)-h(t))/(u-t)\geq 0 \] for all \(a\leq s<t<u\leq b\).
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 28A50
zbMATH DE Number: 4146317
zbMATH Keywords: refinement integral
zbMATH Keywords: Bochner-Radon-Nikodým theorem
zbMATH Keywords: set-valued set- functions
zbMATH Keywords: bounded concavity condition
reviewed by: J. W. Hagood
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4146317