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 $\lambda^*$ denote the Lebesgue outer measure on $\mathbb {R} ^{n}$, and let ${\displaystyle E\subseteq \mathbb {R} ^{n}}$. Then $E$ 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 $A$ 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 $\mathbb {R}$, 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 $\mu ^{*}$ be an outer measure on a set $X$. Then ${\displaystyle E\subset X}$ is called $\mu ^{*}$–measurable if for every $A\subset X$, the equality ${\displaystyle \mu ^{*}(A)=\mu ^{*}(A\cap E)+\mu ^{*}(A\cap E^{\mathrm {c} })}$ holds.

See also

• Carathéodory's theorem (disambiguation)