The Study of Some Questions of LOEB Measure

Abstract: In this paper, many concepts and propositions in measure theory are described and characterized by nonstandard analysis in nonstandard saturation model, and some important conclusions in Loeb measure space are extended and developed to a certain extent. Thus, not only the content of nonstandard analysis has been rich, but also the measure theory has been researched in the view of nonstandard analysis.In the first part, firstly, we introduce the nonstandard analysis theory simply. Using the axioms in nonstandard analysis, the nonstandard model is axiomatic approached, and the existence of nonstandard model and the consistence of axioms in nonstandard analysis are proved by the construction of nonstandard model. Secondly, some properties of nonstandard model are discussed, such as transitivity, Boolean properties, etc. Finally, Loeb measure space(Y,L((?)),v_L) has been constructed by two kind of different methods in internal measure space (Y, (?),v), and their consistency has been proved. Then, some fundamental properties of Loeb measurable function and Loeb integration are discussed.In the second part, we give the definition of Loeb measure space ofσ- finite measure space, discuss its properties; Then the Loeb measure space of image measure has been constructed; Finally, the definition of Loeb counting measure is given, by which, a construction of Lebesgue measure has been given, and discuss some simple properties of Lebesgue measurable and integrable function.Through the research of this paper, on the one hand, it makes the theory and the practice a more perfect union, and enhances us to get more widespread understanding to the subject about nonstandard analysis. On the other hand, it makes the theory of nonstandard analysis richer, and is helpful to the development of nonstandard analysis in other applied domains…
Key words: Loeb measure; σ- finite; image measure; Lebesgue measure

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