Lebesgue differentiation theorem

The Lebesgue differentiation theorem states that for a locally integrable function, averages over shrinking balls converge to the function’s value at almost every point.

Lebesgue differentiation basis: A family of sets whose shrinking averages recover locally integrable functions almost everywhere. Euclidean balls are the standard basis in the theorem’s statement.
Locally integrable function: A measurable function whose absolute value has finite integral on every compact subset of its domain. This is the function class to which the theorem applies.
Fundamental theorem of calculus for Lebesgue integrals: A theorem stating that the indefinite integral of an integrable function is absolutely continuous and has that function as derivative almost everywhere. Lebesgue differentiation identifies the derivative of an indefinite integral almost everywhere.
Pointwise convergence: A sequence of functions converges pointwise when its values converge at each point in a specified domain. The theorem concerns convergence of local averages, not a sequence of functions at every point.
Hardy–Littlewood maximal function: A function assigning each point the supremum of averages of an absolute-value function over neighborhoods containing that point. Its weak-type bound controls where averaging errors can remain large.
Lebesgue measure: A measure on Euclidean space that extends length, area, and volume to broad classes of sets. The theorem measures exceptional points and normalizes ball averages using Lebesgue measure.
Radon–Nikodym theorem: A theorem representing an absolutely continuous measure as an integral against a measurable density. Differentiation of measures recovers the Radon–Nikodym density from local ratios.
Uniform convergence: A sequence of functions converges uniformly when one error bound works across the entire domain. Differentiation holds almost everywhere, not uniformly over all points.
Weak-type inequality: An estimate bounding the measure of a function’s large-value set by an input norm and a threshold. The maximal function’s weak-type estimate is a key step in the proof.
Almost everywhere: A property holds almost everywhere when the set where it fails has measure zero. The theorem allows a measure-zero set of points where differentiation may fail.

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