Carathéodory's Criterion

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.

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. 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.

• 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