MEASURABLE SET-VALUED FUNCTIONS AND BOCHNER INTEGRALS

MEASURABLE SET-VALUED FUNCTIONS AND BOCHNER INTEGRALS

ABSTRACT

This thesis explores the measurability of set-valued functions and the Bochner integral by combining concepts from Topology, Measure Theory, Probability Theory, and Functional Analysis. The study begins with a comprehensive examination of the Hausdorff metric, focusing on its properties and topology. The analysis distinguishes between the cases where E is a metric space and where E is a normed linear space. After an overview of essential theorems, the thesis presents four types of convergence associated with the Hausdorff metric: Hausdorff convergence, Wisjman convergence, Weak convergence, and Kuratowski-Mosco convergence. These convergence notions are subsequently compared.

Additionally, the thesis investigates set-valued random variables and their properties. Five different measures of set-valued functions are studied and compared, alongside two forms of the Bochner integral: the Banach-valued Bochner integral and the set-valued Bochner integral. Through this examination, the thesis contributes to the understanding and analysis of the measurability of set-valued functions and the application of the Bochner integral.

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