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

$\mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}\left(S\cap M^{c}\right)$

holds where $M^{c}:=\Omega \setminus M$  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.