The Metric Integral of Set-Valued Functions

Nira Dyn; Elza Farkhi; Alona Mokhov

doi:10.1007/s11228-017-0403-1

The Metric Integral of Set-Valued Functions

Nira Dyn, Elza Farkhi^*, Alona Mokhov

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in ℝ^d. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.

Original language	English
Pages (from-to)	867-885
Number of pages	19
Journal	Set-Valued and Variational Analysis
Volume	26
Issue number	4
DOIs	https://doi.org/10.1007/s11228-017-0403-1
State	Published - 1 Dec 2018

Keywords

Aumann integral
Compact sets
Kuratowski upper limit
Metric integral
Metric linear combinations
Metric selections
Set-valued functions

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10.1007/s11228-017-0403-1

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abstract = "This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in ℝd. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.",

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AU - Mokhov, Alona

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N2 - This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in ℝd. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.

AB - This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in ℝd. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.

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KW - Compact sets

KW - Kuratowski upper limit

KW - Metric integral

KW - Metric linear combinations

KW - Metric selections

KW - Set-valued functions

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JO - Set-Valued and Variational Analysis

JF - Set-Valued and Variational Analysis

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The Metric Integral of Set-Valued Functions

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