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 [math]\displaystyle{ \lambda^* : {\mathcal P}(\R^n) \to [0, \infty] }[/math] denote the Lebesgue outer measure on [math]\displaystyle{ \R^n, }[/math] where [math]\displaystyle{ {\mathcal P}(\R^n) }[/math] denotes the power set of [math]\displaystyle{ \R^n, }[/math] and let [math]\displaystyle{ M \subseteq \R^n. }[/math] Then [math]\displaystyle{ M }[/math] is Lebesgue measurable if and only if [math]\displaystyle{ \lambda^*(S) = \lambda^*(S \cap M) + \lambda^*\left(S \cap M^c\right) }[/math] for every [math]\displaystyle{ S \subseteq \R^n, }[/math] where [math]\displaystyle{ M^c }[/math] denotes the complement of [math]\displaystyle{ M. }[/math] Notice that [math]\displaystyle{ S }[/math] 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 [math]\displaystyle{ \R, }[/math] 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 [math]\displaystyle{ \mu^* : {\mathcal P}(\Omega) \to [0, \infty] }[/math] is an outer measure on a set [math]\displaystyle{ \Omega, }[/math] where [math]\displaystyle{ {\mathcal P}(\Omega) }[/math] denotes the power set of [math]\displaystyle{ \Omega, }[/math] then a subset [math]\displaystyle{ M \subseteq \Omega }[/math] is called [math]\displaystyle{ \mu^* }[/math]–measurable or Carathéodory-measurable if for every [math]\displaystyle{ S \subseteq \Omega, }[/math] the equality[math]\displaystyle{ \mu^*(S) = \mu^*(S \cap M) + \mu^*\left(S \cap M^c\right) }[/math]holds where [math]\displaystyle{ M^c := \Omega \setminus M }[/math] is the complement of [math]\displaystyle{ M. }[/math]

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

See also

• Carathéodory's extension theorem – Theorem extending pre-measures to measures
• Non-measurable set – Set which cannot be assigned a meaningful "volume"
• Outer measure – Mathematical function
• Vitali set – Set of real numbers that is not Lebesgue measurable