Lebesgue differentiation theorem

E451526

result in measure theory theorem in real analysis

The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.

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Observed surface forms (2)

Surface form	Occurrences
Lebesgue density theorem	1
Lebesgue’s differentiation theorem	1

Statements (48)

Predicate	Object
instanceOf	result in measure theory ⓘ theorem in real analysis ⓘ
appliesTo	L^1 functions ⓘ L^p functions for 1 ≤ p ≤ ∞ ⓘ absolutely continuous measures with respect to Lebesgue measure ⓘ
assumption	function is locally integrable ⓘ underlying measure is a Lebesgue measure or a suitable Radon measure ⓘ
concerns	Lebesgue integrable functions ⓘ Lebesgue measure NERFINISHED ⓘ locally integrable functions ⓘ
conclusion	failure set has measure zero ⓘ pointwise convergence of local averages almost everywhere ⓘ
domain	Euclidean space R^n NERFINISHED ⓘ
field	measure theory ⓘ real analysis ⓘ
generalizationOf	Lebesgue density theorem for measurable sets NERFINISHED ⓘ
hasConsequence	a function is determined almost everywhere by its integrals over balls ⓘ almost everywhere existence of approximate limits of integrable functions ⓘ
hasVariant	differentiation theorem for Radon measures ⓘ differentiation theorem for metric measure spaces ⓘ
historicalPeriod	early 20th century ⓘ
implies	almost everywhere pointwise recovery of a function from its local averages ⓘ
isFundamentalIn	modern integration theory ⓘ the theory of L^p spaces ⓘ
namedAfter	Henri Lebesgue NERFINISHED ⓘ
relatedTo	Fundamental theorem of calculus NERFINISHED ⓘ Hardy–Littlewood maximal theorem NERFINISHED ⓘ Radon–Nikodym theorem NERFINISHED ⓘ martingale convergence theorem NERFINISHED ⓘ
requires	completeness of Lebesgue measure ⓘ
robustUnder	choice of reasonable differentiation bases such as balls or cubes ⓘ
statement	For an L^1_loc function on R^n, the averages over balls shrinking to a point converge almost everywhere to the function value at that point ⓘ If f is locally integrable on R^n, then for almost every x, the limit as r→0 of (1/\|B(x,r)\|)∫_{B(x,r)} f(y) dy equals f(x) ⓘ The set of points where the differentiation formula fails has Lebesgue measure zero ⓘ
typeOfLimit	almost everywhere limit ⓘ
typicalNeighborhoods	balls in Euclidean metric ⓘ cubes in Euclidean space ⓘ
usedIn	ergodic theory ⓘ functional analysis ⓘ harmonic analysis ⓘ partial differential equations ⓘ probability theory ⓘ
usesConcept	Hardy–Littlewood maximal function NERFINISHED ⓘ Lebesgue integral NERFINISHED ⓘ Lebesgue measure zero set NERFINISHED ⓘ Vitali covering theorem NERFINISHED ⓘ density theorem ⓘ maximal inequality ⓘ

Referenced by (5)

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

Hardy–Littlewood maximal function → usedFor → Lebesgue differentiation theorem ⓘ

Hardy–Littlewood maximal function → relatedTo → Lebesgue differentiation theorem ⓘ

Henri Lebesgue → notableConcept → Lebesgue differentiation theorem ⓘ

this entity surface form: Lebesgue’s differentiation theorem

Singular Integrals and Differentiability Properties of Functions → topic → Lebesgue differentiation theorem ⓘ

Steinhaus theorem → relatedTo → Lebesgue differentiation theorem ⓘ

this entity surface form: Lebesgue density theorem