Carathéodory's criterion is a result in

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

that was formulated by Greek

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...

that characterizes when a set is

Lebesgue measurable In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

Statement

Carathéodory's criterion: Let

\lambda^* : \wp(\R^n) \to, \infty /math> denote the

Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...

outer measure In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...

\R^n,

where

\wp(\R^n)

denotes the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

\R^n,

and let

M \subseteq \R^n.

Then

M

if and only if

\lambda^*(S) = \lambda^* (S \cap M) + \lambda^* (S \cap M^\mathrm)

for every

S \subseteq \R^n,

where

M^\mathrm

denotes the

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

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^* : \wp(\Omega) \to, \infty /math> is an

on a set

\Omega,

where

\wp(\Omega)

denotes the

\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^*(S \cap M^\mathrm)

holds where

M^\mathrm := \Omega \setminus M

is the

M.

The family of all

\mu^*

–measurable subsets is a

σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...

(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 Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...

of the outer measure

\mu^*

to this family is a

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

Statement

Generalization

See also