CHAPTER 14 – Set-Valued Integration and Set-Valued Probability Theory: An Overview

This chapter presents an overview of set-valued integration and set-valued probability theory. The mathematical modelization based on multifunctions has shown its great adaptability and relevance for a long time. The integration of strongly measurable multifunctions with those that can be approximated by simple measurable multifunctions in the sense of the Hausdorff distance is analyzed in the chapter. The weakly measurable multifunctions in connection with graph-measurability are presented in the chapter. The Hiai–Umegaki set-valued conditional expectation is examined and set-valued measures are considered in the chapter. The definition and the main properties of set-valued martingales and some basic facts about Gaussian multifunctions and the set-valued central limit theorem are presented in the chapter. The integration of strongly measurable multifunctions is elaborated in the chapter. It allows for a lot of stability properties and for general results of existence of measurable selections, especially in connection with the graph measurability. It is shown in the chapter that the measurability of a closed-valued multifunction can be expressed in terms of distance functions and is connected with the notion of Castaing representation.

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