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.

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.

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 \(\mu^*\)–measurable or Carathéodory-measurable 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.