Carathéodory's criterion explained

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

λ ^{*} : { l P}(\R ^{n)} \to [ 0 , inf ty] denote the Lebesgue outer measure on \R ^{n,} where { l 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 λ ^{*(S)} = λ ^{*(S} \cap M) + λ ^{*\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.^[1]

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 ^{*} : { l P}(\Omega) \to [ 0 , inf ty] is an outer measure on a set \Omega, where { l 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$$\backslash mu^*(S)\; =\; \backslash mu^*(S\; \backslash cap\; M)\; +\; \backslash mu^*\backslash left(S\; \backslash cap\; M^c\backslash right)$$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.

Notes and References

1. Book: Pugh, Charles C. . Real Mathematical Analysis . Springer . 978-3-319-17770-0 . 2nd . 388 . en.