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

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A locally integrable function on $R^{n}$

Definition

Let $f : R^{n} \to C$ be a measurable function (Borel measurable and Lebesgue measurable functions on $R^{n}$ ). We say that $f$ is locally integrable on $R^{n}$ if for every Euclidean ball $B (x, r)$ with $r > 0$ (Open ball, closed ball and sphere in a metric space) one has $\int_{B (x, r)} ∣ f (y) ∣ d λ (y) < \infty,$ so the restriction of $f$ to each ball is integrable in the sense of Integrable real and complex functions, and their integrals.

The set of such functions is denoted by $L_{l o c}^{1} (R^{n})$ . When convenient, the same notation is also used for the corresponding almost-everywhere equivalence classes.

Depends on

Used by

Lebesgue points and the Lebesgue set of an L¹_loc class Definition
The average of a locally integrable function over a Euclidean ball Definition
Lebesgue differentiation theorem on ℝⁿ Theorem

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Sources

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Section 3.4 (standard reference, not scraped)
Terence Tao, An Introduction to Measure Theory, Exercise 1.6.14 (standard reference, not scraped)

A locally integrable function on RnRn