Lebesgue density theorem

For every Lebesgue-measurable set in Euclidean space, the fraction of a sufficiently small ball occupied by the set tends to one at almost every point inside it and zero at almost every point outside it.

Lebesgue differentiation theorem: A theorem stating that local averages of an integrable function converge to its value almost everywhere. Applying it to a set's indicator function yields the density conclusion.
Lebesgue-measurable set: A subset of Euclidean space whose indicator function is measurable with respect to Lebesgue measure. The theorem applies to these sets, including sets that differ from Borel sets by null sets.
Density point: A point where the proportion of a measurable set in shrinking neighborhoods tends to one. Almost every point of the set is a density point by this theorem.
Topological boundary: The boundary of a set is the closure of the set minus its interior. Topological boundary membership alone does not determine the measure-theoretic density at a point.
Hardy–Littlewood maximal function: An operator assigning each point the supremum of local averages of a function over neighborhoods containing it. Its weak-type bound controls points where small-scale averages deviate persistently.
Almost everywhere: A property holds almost everywhere when the points where it fails form a set of measure zero. Density can fail at individual points, but only on a null set.
Lebesgue point: A point where the mean local deviation of a function from its value tends to zero. Density points are the indicator-function instance of this broader notion.
Essential boundary: The measure-theoretic boundary consists of points where both a set and its complement occupy positive limiting proportions. It captures boundary behavior through densities rather than topology alone.
Vitali covering theorem: A covering theorem that selects disjoint sets from a fine family while covering the original set up to negligible measure. Covering arguments underpin maximal inequalities used to prove density results.
Euclidean ball: The set of points within a fixed distance of a center in Euclidean space. The theorem measures local proportions using balls centered at each point.

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Denjoy–Luzin theoremRelated: Density points turn measure-theoretic control of value sets into approximate continuity.