The Metric Integral of Set-Valued Functions

Nira Dyn, Elza Farkhi*, Alona Mokhov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)867-885
Number of pages19
JournalSet-Valued and Variational Analysis
Issue number4
StatePublished - 1 Dec 2018


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


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