Carathéodory's criterion

Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable.

Statement

Carathéodory's criterion: Let ${\lambda }^{\ast }:𝒫({ℝ}^n)\to [0,\mathrm{\infty }]$ denote the Lebesgue outer measure on ${ℝ}^n,$ where $𝒫({ℝ}^n)$ denotes the power set of ${ℝ}^n,$ and let $M\subseteq {ℝ}^n.$ Then $M$ is Lebesgue measurable if and only if ${\lambda }^{\ast }(S)={\lambda }^{\ast }(S\cap M)+{\lambda }^{\ast }(S\cap M^c)$ for every $S\subseteq {ℝ}^n,$ where $M^c$ denotes the complement of $M.$ Notice that $S$ is not required to be a measurable set.

Generalization

The Carathéodory criterion is of considerable importance because, in contrast to Lebesgue's original formulation of measurability, which relies on certain topological properties of $ℝ,$ this criterion readily generalizes to a characterization of measurability in abstract spaces. Indeed, in the generalization to abstract measures, this theorem is sometimes extended to a definition of measurability. Thus, we have the following definition: If ${\mu }^{\ast }:𝒫(\mathrm{\Omega })\to [0,\mathrm{\infty }]$ is an outer measure on a set $\mathrm{\Omega },$ where $𝒫(\mathrm{\Omega })$ denotes the power set of $\mathrm{\Omega },$ then a subset $M\subseteq \mathrm{\Omega }$ is called Template:Em or Template:Em if for every $S\subseteq \mathrm{\Omega },$ the equality$${\mu }^{\ast }(S)={\mu }^{\ast }(S\cap M)+{\mu }^{\ast }(S\cap M^c)$$holds where $M^c:=\mathrm{\Omega }\setminus M$ is the complement of $M.$

The family of all ${\mu }^{\ast }$–measurable subsets is a σ-algebra (so for instance, the complement of a ${\mu }^{\ast }$–measurable set is ${\mu }^{\ast }$–measurable, and the same is true of countable intersections and unions of ${\mu }^{\ast }$–measurable sets) and the restriction of the outer measure ${\mu }^{\ast }$ to this family is a measure.

See also

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References

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