DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-04

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Density of a measurable set at a point

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $E \subseteq R^{n}$ be Lebesgue measurable (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ), and let $x \in R^{n}$ . If the limit exists, the density of $E$ at $x$ is $Θ (E, x) : = \lim_{r \to 0^{+}} \frac{λ (E \cap B (x, r))}{λ (B (x, r))},$ where the denominator is positive and finite by Euclidean balls have positive finite Lebesgue measure.

When $Θ (E, x) = 1$ , we call $x$ a density-one point of $E$ ; when $Θ (E, x) = 0$ , we call $x$ a density-zero point of $E$ .

Depends on

Used by

An endpoint of an interval has density one half, not one Example
Lebesgue density theorem Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Terence Tao, An Introduction to Measure Theory, Exercise 1.6.24 (standard reference, not scraped)
Walter Rudin, Real and Complex Analysis, 3rd ed., Chapter 7 (standard reference, not scraped)