Measurable Set-Valued Functions and Bochner Integrals

Abstract

In this thesis, several concepts from Topology, Measure Theory, Probability Theory, and Functional analysis were combined in the study of the measurability of set-valued functions and the Bochner integral. We started with a detailed study of the Hausdorff metric, its properties, and topology by exposing separately the case where E is a metric space and the case where E is a normed linear space. After reviewing the important theorems, we present the four convergences related to Hausdorff metric: Hausdorff convergence, Wisjman convergence, Weak convergence, and Kuratowski-Mosco convergence; and then compared them. Further, set-valued random variables and their properties were studied. We study and compare five types of measures of set-valued functions and the two forms of Bochner integral, that is, the Banach-valued and set-valued Bochner integrals.