Carathéodory measurability criterion
E506846
The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in measure theory
ⓘ
mathematical criterion ⓘ |
| appliesTo | outer measure spaces ⓘ |
| assumes |
monotonicity of the outer measure
ⓘ
null empty set for the outer measure ⓘ subadditivity of the outer measure ⓘ |
| basedOn | Carathéodory construction of measures NERFINISHED ⓘ |
| category | theorem in analysis ⓘ |
| characterizes | measurable sets ⓘ |
| concerns | structure of measurable sets relative to an outer measure ⓘ |
| contrastsWith |
definitions of measurability via Borel σ-algebra
ⓘ
definitions of measurability via open sets ⓘ |
| coreCondition |
Carathéodory condition for measurability
ⓘ
additivity over disjoint unions ⓘ additivity over intersections and complements ⓘ |
| defines | Carathéodory-measurable set ⓘ |
| ensures |
compatibility of measure with set operations
ⓘ
measurable sets form a σ-algebra ⓘ restriction of outer measure to measurable sets is a measure ⓘ |
| expressedIn | set-theoretic language ⓘ |
| field |
measure theory
ⓘ
real analysis ⓘ |
| formalStatement | A set E is measurable if for every set A, μ*(A) = μ*(A ∩ E) + μ*(A \ E) ⓘ |
| guarantees | completeness of the resulting measure ⓘ |
| hasDomain | subsets of a given set X ⓘ |
| holdsIn |
abstract measure spaces
ⓘ
metric measure spaces ⓘ |
| implies |
countable additivity on the σ-algebra of measurable sets
ⓘ
σ-additivity of the induced measure on measurable sets ⓘ |
| introducedBy | Constantin Carathéodory NERFINISHED ⓘ |
| involves | outer measure μ* ⓘ |
| mathematicalSubjectClassification | 28A12 ⓘ |
| namedAfter | Constantin Carathéodory NERFINISHED ⓘ |
| relatedTo |
Carathéodory extension theorem
NERFINISHED
ⓘ
Lebesgue outer measure NERFINISHED ⓘ σ-algebra of measurable sets ⓘ |
| requires | all subsets A of the underlying space ⓘ |
| timePeriod | early 20th century ⓘ |
| typeOf | necessary and sufficient condition for measurability ⓘ |
| usedFor |
constructing measures from outer measures
ⓘ
defining Lebesgue measurable sets ⓘ extending premeasures to complete measures ⓘ |
| usedIn |
construction of Lebesgue measure on ℝ
ⓘ
construction of product measures ⓘ integration theory ⓘ probability theory ⓘ |
| usesConcept | outer measure ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.