Loeb space

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In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using non-standard analysis.

Construction[edit]

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.

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