Carathéodory's criterion

Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory. Its statement is as follows: Let ${\displaystyle \lambda ^{*}}$ denote the Lebesgue outer measure on ${\displaystyle \mathbb {R} ^{n}}$, and let ${\displaystyle E\subseteq \mathbb {R} ^{n}}$. Then ${\displaystyle E}$ is Lebesgue measurable if and only if ${\displaystyle \lambda ^{*}(A)=\lambda ^{*}(A\cap E)+\lambda ^{*}(A\cap E^{\mathrm {c} })}$ for every ${\displaystyle A\subseteq \mathbb {R} ^{n}}$. Notice that ${\displaystyle A}$ is not required to be a measurable set.

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. Thus, we have the following definition: Let ${\displaystyle \mu ^{*}}$ be an outer measure on a set ${\displaystyle X}$. Then ${\displaystyle E\subset X}$ is called ${\displaystyle \mu ^{*}}$–measurable if for every ${\displaystyle A\subset X}$, the equality ${\displaystyle \mu ^{*}(A)=\mu ^{*}(A\cap E)+\mu ^{*}(A\cap E^{\mathrm {c} })}$ holds.

See also

• Carathéodory's theorem (disambiguation)