The Auman Integral of Set-Valued Maps

Abstract

This thesis focuses on the Aumann integral of set-valued random variables and its properties. We started o by studying the space in which this integral lies: hyperspace endowed with the Hausdor metric. We considered convergence on a hyperspace with respect to the Hausdor metric and reviewed the works of Kuratowski, Mosco in trying to abstract topologically, the Hausdor convergence; this led to a comparison between weak, Wijsmann, Kuratowski-Mosco convergences to Hausdor convergence. We proceeded to see the conditions under which a set-valued random variable is measurable, integrable and integrably bounded. Finally, we de ned the class of integrable selections of an integrable set-valued random variable and used it to de ne the Aumann integral, and went further to prove and outline su cient conditions for the Aumann integral to be convex and closed-valued respectively