Carathéodory's extension theorem

Carathéodory's extension theorem states that a premeasure on an algebra of sets extends to a measure on the generated σ-algebra. The extension is unique when the premeasure is σ-finite.

Premeasure: A nonnegative, countably additive set function defined on an algebra, with countable additivity applying when the union remains in the algebra. This is the initial set function whose values the theorem extends.
Algebra of sets: A family of subsets containing the whole space and closed under complements and finite unions. The premeasure is defined on this smaller family before extension.
Lebesgue measure: The translation-invariant measure on the real line and its higher-dimensional analogues, extending interval length and volume. Its construction extends interval lengths from a simple algebra to a σ-algebra of measurable sets.
Hahn–Kolmogorov theorem: An extension theorem asserting that a σ-finite premeasure on a ring extends uniquely to the generated σ-algebra. It is a closely related formulation emphasizing extension from a ring rather than an algebra.
Outer measure: A monotone, countably subadditive set function defined on every subset of a space and equal to zero on the empty set. The extension is built by covering arbitrary sets and taking the infimum of premeasure sums.
σ-algebra: A family of subsets containing the whole space and closed under complements and countable unions. This is the larger domain on which the extended measure is defined.
Lebesgue–Stieltjes measure: A measure on the real line associated with a nondecreasing, suitably normalized right-continuous function. A distribution function specifies interval values that can be extended to a measure.
Daniell–Stone theorem: An extension theorem that constructs measures from positive linear functionals on suitable spaces of functions. It builds measures from functionals instead of starting with set values on an algebra.
Carathéodory measurability criterion: A condition requiring a set to split every test set's outer measure additively into its parts inside and outside. It identifies which sets support a genuine measure after the outer measure is constructed.
Measure: A countably additive, nonnegative set function on a σ-algebra that assigns zero to the empty set. The theorem's conclusion is a measure, not merely an extension as a set function.

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