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^* : (\R^n) \to , \infty/math> denote the Lebesgue outer measure on $\R^n,$ where $(\R^n)$ 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^* : (\Omega) \to , \infty/math> is an outer measure on a set $\Omega,$ where $(\Omega)$ 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.

See also

*
*
*
*
*

References

Measure theory

{{probability-stub