Discussions of Some Problems of the Loeb Measure’s Absolute Continuity and Singularity

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. On one hand, the inherent characteristics of nonstandard measure theory have been reflected through discussing internal measure space and Loeb measure space. On the other hand, a kind of method has been offered for studying measure theory further by nonstandard analysis.In the first and second chapter, the nonstandard analysis theory is briefly introduced .Using the axioms in nonstandard analysis, the nonstandard analysis is axiomatic approached. On this basis, some properties of nonstandard model and nonstandard saturation model are discussed. And several equivalent conditions of saturation are given.In chapter three, Loeb measure space(Y,L(A),v_L) has been constructed by two kind of different methods in internal measure space(Y,A,v), and their consistencyhas been proved. Then, some fundamental properties of Loeb measurable function and Loeb integration are discussed.In chapter four, the definition of the absolute continuity of Loeb measure has been given at first, and discussed the properties of the absolute continuity of Loeb measure, also, the relation between the internal finitely additive measure space and the Loeb measure space has been given. Then, the Radon-Nikodym theorem on the Loeb measure space is discussed.In the last chapter, the definition of the singularity of the internal finitely additive measure space and the Loeb measure space态the definition of support have been given. Then, the Lebesgue decomposition of the internal finitely additive measure space and the Loeb measure space are discussed. Finally, the important result: L(~*v) = L(~*v_a) + L(~*v_s) is given…
Key words: Loeb measure; Absolute Continuity; Singularity; Lebesgue decomposition

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