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In this paper,the universal Loeb-measurability definetion of finite measure,σ-finite measures and product measures are given inκ-saturated nonstandard models.And it is shown that many important external sets of nonstandard analysis are universally Loeb-measurable.This extends some important conclusions of Loeb space to some degrees.In Chapter 1 and Chapter 2 of this paper,the background,the actual development condition and some basic theories of nonstandard analysis are presented.In addition, some basic properties ofκ-saturated nonstandard models and nonstandard universe are discussed,and several conditions of equivalence forκ-saturated nonstandard models are given.In Chapter 3,Loeb measure space(X,L(B),L(ν)) has been constructed by two kind of different methods in internal measure space(X,,B,ν),and their consistency has been proved.In Chapter 4,the universal Loeb-measurability of finite measure is given,and then, discussed many important external sets of nonstandard analysis—such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points—are universally Loeb-measurable.,and discussed their property.Moreover,the expressions forτ-smooth Baire-measures andτ-smooth Borel-measures are presented.Finally,a definition for the universal Loeb-measurability ofσ-finite measures is provided.In Chapter 5,the definitions of the universal Loeb-measurability of finite measure product space andσ-finite measure product space,together with some of their universally Loeb-measurable sets are provided.
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