A Characterization Theorem for Aumann Integrals

TL;DR

This paper proves a characterization theorem for Aumann integrals of set-valued measurable functions, extending classical results to the set-valued context with new geometric conditions.

Contribution

It establishes necessary and sufficient conditions for a functional to be represented as an Aumann integral of set-valued functions, generalizing the Daniell-Stone theorem.

Findings

Aumann integral of closed convex upper set-valued functions is itself a closed convex upper set.

The theorem provides conditions including conlinearity, monotone convergence, and geometric properties.

The results extend classical integral representation theorems to the set-valued setting.

Abstract

A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some recent developments in set-valued variational analysis and set optimization. It is shown that the Aumann integral of such a function is also a closed convex upper set. The main theorem characterizes the conditions under which a functional that maps from a certain collection of measurable set-valued functions into the set of all closed convex upper sets can be written as the Aumann integral with respect to some $\sigma$-finite measure. These conditions include the analog of the conlinearity and monotone convergence properties of the classical Daniell-Stone theorem for the Lebesgue integral, and three geometric properties that are peculiar to the set-valued case as they are redundant in the one-dimensional setting.