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

Statement

Generalization

See also