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Loeb space
In mathematics, a Loeb space is a type of measure space introduced by using non-standard analysis.
==Construction==
Loeb's construction starts with a finitely additive map ν from an internal algebra ''A'' of sets to the non-standard reals. Define μ to be given by the standard part of ν, so that μ is a finitely additive map from ''A'' to the extended reals R∪∞∪–∞. Even if ''A'' is a non-standard σ-algebra, the algebra ''A'' need not be an ordinary σ-algebra as it is not usually closed under countable unions. Instead the algebra ''A'' has the property that if a set in it is the union of a countable family of elements of ''A'', then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as μ) from ''A'' to the extended reals is automatically countably additive. Define ''M'' to be the σ-algebra generated by ''A''. Then by Carathéodory's extension theorem the measure μ on ''A'' extends to a countably additive measure on ''M'', called a Loeb measure.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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