Carathéodory's criterion

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

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