Lebesgue measurable set

E898511

mathematical concept measurable set subset of Euclidean space

A Lebesgue measurable set is a subset of Euclidean space for which a consistent, translation-invariant notion of "size" (Lebesgue measure) can be assigned, forming the foundation of modern measure theory and integration.

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All labels observed (1)

Label	Occurrences
Lebesgue measurable set canonical	1

Statements (46)

Predicate	Object
instanceOf	mathematical concept ⓘ measurable set ⓘ subset of Euclidean space ⓘ
associatedWith	Lebesgue measure NERFINISHED ⓘ
characterizedBy	Carathéodory measurability condition ⓘ equality of outer measure of set and outer measure of its intersection with any other set plus that of its complement intersection ⓘ
contrastedWith	non-measurable set ⓘ
definedOn	Euclidean space ⓘ ℝ ⓘ ℝⁿ ⓘ
exampleOf	measurable subset of a measure space ⓘ
fieldOfStudy	integration theory ⓘ measure theory ⓘ real analysis ⓘ
foundationFor	Lᵖ spaces NERFINISHED ⓘ modern probability theory ⓘ modern real analysis ⓘ
generalizationOf	Jordan measurable set ⓘ
hasCardinalityProperty	collection has cardinality 2^{continuum} ⓘ
hasLimitation	not every subset of ℝ is Lebesgue measurable ⓘ
hasProperty	almost-everywhere equality defined via Lebesgue measurable sets ⓘ closed under complementation ⓘ closed under countable intersections ⓘ closed under countable unions ⓘ completeness under Lebesgue measure ⓘ countable additivity of measure ⓘ every Borel set is Lebesgue measurable ⓘ forms a σ-algebra ⓘ inner regularity with respect to closed sets ⓘ measure zero sets are Lebesgue measurable ⓘ outer regularity with respect to open sets ⓘ translation invariance of measure ⓘ
namedAfter	Henri Lebesgue NERFINISHED ⓘ
relatedConcept	Borel set ⓘ Lebesgue integral NERFINISHED ⓘ completion of a measure space ⓘ measurable function ⓘ null set ⓘ outer measure ⓘ σ-algebra ⓘ
subsetOf	power set of ℝ ⓘ σ-algebra of Lebesgue measurable subsets of ℝ ⓘ
usedFor	definition of Lebesgue integral ⓘ definition of measurable functions ⓘ formulation of convergence theorems in integration ⓘ probability spaces on ℝⁿ ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Henri Lebesgue → notableConcept → Lebesgue measurable set ⓘ