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 ∗ : P ( R n ) → [ 0 , ∞ ] {\displaystyle \lambda ^{*}:{\mathcal {P}}(\mathbb {R} ^{n})\to [0,\infty ]} denote the Lebesgue outer measure on R n , {\displaystyle \mathbb {R} ^{n},} where P ( R n ) {\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})} denotes the power set of R n , {\displaystyle \mathbb {R} ^{n},} and let M ⊆ R n . {\displaystyle M\subseteq \mathbb {R} ^{n}.} Then M {\displaystyle M} is Lebesgue measurable if and only if λ ∗ ( S ) = λ ∗ ( S ∩ M ) + λ ∗ ( S ∩ M c ) {\displaystyle \lambda ^{*}(S)=\lambda ^{*}(S\cap M)+\lambda ^{*}\left(S\cap M^{c}\right)} for every S ⊆ R n , {\displaystyle S\subseteq \mathbb {R} ^{n},} where M c {\displaystyle M^{c}} denotes the complement of M . {\displaystyle M.}
Notice that S {\displaystyle 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 , {\displaystyle \mathbb {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 μ ∗ : P ( Ω ) → [ 0 , ∞ ] {\displaystyle \mu ^{*}:{\mathcal {P}}(\Omega )\to [0,\infty ]} is an outer measure on a set Ω , {\displaystyle \Omega ,} where P ( Ω ) {\displaystyle {\mathcal {P}}(\Omega )} denotes the power set of Ω , {\displaystyle \Omega ,} then a subset M ⊆ Ω {\displaystyle M\subseteq \Omega } is called μ ∗ {\displaystyle \mu ^{*}} –measurable or Carathéodory-measurable if for every S ⊆ Ω , {\displaystyle S\subseteq \Omega ,} the equality μ ∗ ( S ) = μ ∗ ( S ∩ M ) + μ ∗ ( S ∩ M c ) {\displaystyle \mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}\left(S\cap M^{c}\right)} holds where M c := Ω ∖ M {\displaystyle M^{c}:=\Omega \setminus M} is the complement of M . {\displaystyle M.}
The family of all μ ∗ {\displaystyle \mu ^{*}} –measurable subsets is a σ-algebra (so for instance, the complement of a μ ∗ {\displaystyle \mu ^{*}} –measurable set is μ ∗ {\displaystyle \mu ^{*}} –measurable, and the same is true of countable intersections and unions of μ ∗ {\displaystyle \mu ^{*}} –measurable sets) and the restriction of the outer measure μ ∗ {\displaystyle \mu ^{*}} to this family is a measure.
See also Carathéodory's extension theorem – Theorem extending pre-measures to measures Non-Borel set – Class of mathematical setsPages displaying short descriptions of redirect targets 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
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