Loeb space

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

Construction

Loeb's construction starts with a finitely additive map $\nu$ from an internal algebra $𝒜$ of sets to the nonstandard reals. Define $\mu$ to be given by the standard part of $\nu$, so that $\mu$ is a finitely additive map from $𝒜$ to the extended reals $\overline{ℝ}$. Even if $𝒜$ is a nonstandard $\sigma$-algebra, the algebra $𝒜$ need not be an ordinary $\sigma$-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 $\mu$) from $𝒜$ to the extended reals is automatically countably additive. Define $ℳ$ to be the $\sigma$-algebra generated by $𝒜$. Then by Carathéodory's extension theorem the measure $\mu$ on $𝒜$ extends to a countably additive measure on $ℳ$, called a Loeb measure.

External links

• Home page of Peter Loeb