The Study of Representation and Some Corresponding Questions of the LOEB Measure

Abstract: In this paper, based on the basic theories of loeb measure, several properties of the inverse standard part map are discussed inκ-saturated nonstandard models, and the definition of outer Loeb measure representation of standard measure. Then the question that which measures can be represented by a partial map is discussed,and some properties of set function are proved. We mainly study some methods of representing measures by Loeb measure.In Chapter 1 and Chapter 2 of this paper, firstly, the background, the development condition and some basic theories of nonstandard analysis are presented simply. Secondly, the relation between standard universe and nonstandard universe is discussed. Thirdly, nonstandard models and its some properties are studied.In Chapter 3, two kinds of different methods for construction of Loeb measure space has been presented, and their consistency has been proved. Based on it, Loeb measurable function and Loeb integration and its basic theories are studied.In Chapter 4, firstly, several properties of the inverse standard part map are discussed; secondly, the definition of outer Loeb measure representation of standard measureμis given, and one of sufficient and necessary conditions that (?) is an outer Loeb measure representation ofμare proved; thirdly, the definition that the measure possesses a loeb representation by a partial map is provided, the question that which measures can be represented by a partial map is discussed; finally, by an exemple we obtain that the set of loeb integrable subsets is not an algebra in general, some properties of internal subadditive and additive set function are discussed. It is proved that (?)st-1 is (?)-smooth Borel measure provided that X is a regular space and (?) is internal, subadditive and additive set function…
Key words: nonstandard model; Loeb measure; σ—algebra; inverse standard part map

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