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 ${\displaystyle \lambda ^{*}:{\mathcal {P}}(\mathbb {R} ^{n})\to [0,\infty ]}$ denote the Lebesgue outer measure on ${\displaystyle \mathbb {R} ^{n},}$ where ${\displaystyle {\mathcal {P}}(\mathbb {R} ^{n})}$ denotes the power set of ${\displaystyle \mathbb {R} ^{n},}$ and let ${\displaystyle M\subseteq \mathbb {R} ^{n}.}$ Then ${\displaystyle M}$ is Lebesgue measurable if and only if ${\displaystyle \lambda ^{*}(S)=\lambda ^{*}(S\cap M)+\lambda ^{*}\left(S\cap M^{c}\right)}$ for every ${\displaystyle S\subseteq \mathbb {R} ^{n},}$ where ${\displaystyle M^{c}}$ denotes the complement of ${\displaystyle M.}$ Notice that ${\displaystyle S}$ 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 ${\displaystyle \mathbb {R} ,}$ 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.^[1] Thus, we have the following definition: If ${\displaystyle \mu ^{*}:{\mathcal {P}}(\Omega )\to [0,\infty ]}$ is an outer measure on a set ${\displaystyle \Omega ,}$ where ${\displaystyle {\mathcal {P}}(\Omega )}$ denotes the power set of ${\displaystyle \Omega ,}$ then a subset ${\displaystyle M\subseteq \Omega }$ is called ${\displaystyle \mu ^{*}}$–measurable or Carathéodory-measurable if for every ${\displaystyle S\subseteq \Omega ,}$ the equality

$${\displaystyle \mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}\left(S\cap M^{c}\right)}$$

${\displaystyle M^{c}:=\Omega \setminus M}$

${\displaystyle M.}$

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

References

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