The Lebesgue Differentiation Theorem states that for almost every point in the domain of a locally integrable function, the average of the function over small balls shrinking to that point converges to the function's value there. Named for Henri Lebesgue, it is a foundational result of real analysis showing that the fundamental theorem of calculus continues to hold, in an averaged almost-everywhere sense, for functions far more general than continuous ones.
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StatementFor a locally integrable function f on R^n, at almost every point x the average of f over a ball B shrinking to x converges to f(x) as the diameter of B tends to zero. Proved by Henri Lebesgue in 1910, extending his own 1904 one-dimensional result. 1 Classification
Statement FormCharacterization Theorem 1 Lebesgue Differentiation Theorem
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Source Lebesgue Differentiation Theorem (Wikipedia)
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1. Lebesgue Differentiation Theorem (Wikipedia)
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The Lebesgue differentiation theorem (Lebesgue 1910) states that this derivative exists and is equal to f(x) at almost every point x ∈ Rn.
Principal Result Of: Henri Lebesgue, Lead sentence of the article on the result
The theorem is named for Henri Lebesgue.
In Branch: Real Analysis, Lead sentence
Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integra
Proved By: Henri Lebesgue, Lead paragraph
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable
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