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 $$\lambda^* : {\mathcal P}(\R^n) \to [0, \infty]$$ denote the Lebesgue outer measure on $$\R^n,$$ where $${\mathcal P}(\R^n)$$ denotes the power set of $$\R^n,$$ and let $$M \subseteq \R^n.$$ Then $$M$$ is Lebesgue measurable if and only if $$\lambda^*(S) = \lambda^*(S \cap M) + \lambda^*\left(S \cap M^c\right)$$ for every $$S \subseteq \R^n,$$ 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 $$\R,$$ 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 ' or ' 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.