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 ${\lambda }^{*}:{𝒫}({\mathbb{ℝ}}^n)\to [0,\mathrm{\infty }]$ denote the Lebesgue outer measure on ${\mathbb{ℝ}}^n,$ where ${𝒫}({\mathbb{ℝ}}^n)$ denotes the power set of ${\mathbb{ℝ}}^n,$ and let $M\subseteq {\mathbb{ℝ}}^n.$ Then $M$ is Lebesgue measurable if and only if ${\lambda }^{*}(S)={\lambda }^{*}(S\cap M)+{\lambda }^{*}(S\cap M^c)$ for every $S\subseteq {\mathbb{ℝ}}^n,$ 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 $\mathbb{ℝ},$ 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 ${\mu }^{*}:{𝒫}(\mathrm{\Omega })\to [0,\mathrm{\infty }]$ 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 ${\mu }^{*}$–measurable or Carathéodory-measurable if for every $S\subseteq \mathrm{\Omega },$ the equality$${\mu }^{*}(S)={\mu }^{*}(S\cap M)+{\mu }^{*}(S\cap M^c)$$holds where $M^c:=\mathrm{\Omega }\setminus M$ is the complement of $M.$

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

See also

• Carathéodory's extension theorem – Theorem extending pre-measures to measures
• Non-measurable set – Set which cannot be assigned a meaningful "volume"
• Outer measure – Mathematical function
• Vitali set – Set of real numbers that is not Lebesgue measurable