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 ${\displaystyle \lambda ^{*}:{\mathcal {P))(\mathbb {R} ^{n})\to [0,\infty ]}$ denote the Lebesgue outer measure on ${\displaystyle \mathbb {R} ^{n},}$ where ${\displaystyle {\mathcal {P))(\mathbb {R} ^{n})}$ denotes the power set of ${\displaystyle \mathbb {R} ^{n},}$ and let ${\displaystyle M\subseteq \mathbb {R} ^{n}.}$ Then ${\displaystyle M}$ is Lebesgue measurable if and only if ${\displaystyle \lambda ^{*}(S)=\lambda ^{*}(S\cap M)+\lambda ^{*}\left(S\cap M^{c}\right)}$ for every ${\displaystyle S\subseteq \mathbb {R} ^{n},}$ where ${\displaystyle M^{c))$ denotes the complement of ${\displaystyle M.}$ Notice that ${\displaystyle S}$ is not required to be a measurable set.^[1]

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 ${\displaystyle \mathbb {R} ,}$ 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.^[1] Thus, we have the following definition: If ${\displaystyle \mu ^{*}:{\mathcal {P))(\Omega )\to [0,\infty ]}$ is an outer measure on a set ${\displaystyle \Omega ,}$ where ${\displaystyle {\mathcal {P))(\Omega )}$ denotes the power set of ${\displaystyle \Omega ,}$ then a subset ${\displaystyle M\subseteq \Omega }$ is called ${\displaystyle \mu ^{*))$–measurable or Carathéodory-measurable if for every ${\displaystyle S\subseteq \Omega ,}$ the equality$${\displaystyle \mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}\left(S\cap M^{c}\right)}$$holds where ${\displaystyle M^{c}:=\Omega \setminus M}$ is the complement of ${\displaystyle M.}$

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