The Decomposition Of Monotone Set-Valued Measures And Properties Of Measurable Functions

This paper mainly studies the properties of measurable functions and the decomposition theorem about atoms on monotone set-valued measure spaces.In the first part,the concepts of propertyS~*,propertyPS~*,the Egoroff condition and conditionM~*are given.The relationships between them are discussed.Based on these,the equivalent conditions to Egoroff type Theorem and two kinds of Riesz type Theorem on monotone set-valued measure spaces are obtained.In the second part,the monotone set-valued measures defined on Borel?-algebra of a topological space is discussed.To begin with,the concepts of inner regular,outer regular and regular of a monotone set-valued measure are presented and the sufficient(necessary)condition for regularity of monotone set-valued measures is investigated.Then by using the continuity from above,a version of Egoroff type Theorem for monotone set-valued measures is given.Furthermore,some related corollaries are shown.And based on these,Lusin type Theorem for monotone set-valued measures is proved.In the third part,by the null-additivity and some other conditions for monotone set-valued measures,some properties of atoms of monotone set-valued measures are discussed.Then we give the concept of minimal atoms of monotone set-valued measures and study the properties of these minimal atoms.Moreover,the concept of pseudo-atoms is introduced and their properties are shown as well.Finally,by using the conclusions which have been obtained,the Saks decomposition theorem and Darboux property for monotone set-valued measures are investigated.