Abstract: In this paper,the nonstandard analysis theory is used for inducing a metric space by a Loeb measure space.On this basis,a metric space is induced by a internal finitely additive measure space.The close relationship between the metric space induced by a Loeb measure space and the metric space induced by a internal finitely additive measure space is illustrated with the concepts and some properties of Loeb measure.Then,some properties of the metric space that induced by a internal finitely additive measure space are studied.In the first two chapters,we first Succinctly present the origin,development and research states of the nonstandard analysis.Then,the theoretical foundation of nonstandard analysis as well as the axiomatic nonstandard analysis are given.Finally, the nonstandard model and the saturation model are discussed,as well as some natures of the nonstandard model and several equivalent conditions of saturation model are given.In the third chapter,two different methods which construct the Loeb measure are researched,and their consistency is proved.Next some basic natures of Loeb measurable function and Loeb integration have been discussed.In the fourth chapter,first of all,a metric space is derived by the Loeb measure space which is a standard measure space,follow that the Completeness,separability and the Baire theorem are expressed.In the fifth chapter,we mainly discuss the relationship of the metric space induced by a Loeb measure space and the metric space induced by a internal finitely additive measure space.Last of all,the natures of the metric space induced by a internal finitely additive measure space are studied…

Key words: Internal finitely additive measure; Loeb measure; metric space; Completeness; separability

# The Study of Some Questions of Metric Space Induced by a Internal Finitely Additive Measure Space

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