Vol. 58, No. 2, 1975

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Integration of compact set-valued functions

Zvi Artstein and John Allen Burns

Vol. 58 (1975), No. 2, 297–307

A theory of integration of compact set-valued functions is provided by applying the McShane φ-integral. This integral is a Riemann-type integral and includes the Bochner, Lebesgue and other types of integrals, and by using Riemann sums it avoids deep measure theory. Thus, the φ-integral of set-valued functions contains other types of integrals such as the Hukuhara and Debreu integrals. Generalizations of known results, including the convexity of the integral, are obtained, and the techniques do not require measure theory. Further, if a set-valued function is φ-integrable, then its integral equals the Aumann integral, where the latter is defined as the collection of integrals of selections.

Mathematical Subject Classification 2000
Primary: 28A45
Secondary: 46G10
Received: 25 February 1974
Published: 1 June 1975
Zvi Artstein
John Allen Burns