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

Weighted-integrable Hᵖ functions have vanishing moments in the atomic range Corollary
Product of two distributions is not canonically defined Counterexample
The fundamental Hessian is not absolutely locally integrable Counterexample
Axis-parallel cubes, their averages, and cube maximal functions Definition
BMO seminorm and the quotient by constants Definition
Bounded BV data on the real line Definition
Calderón–Zygmund kernels and their associated operators Definition
Distributional weak solutions of the Cauchy problem Definition
Hᵖ atoms with a prescribed moment order Definition
Kruzhkov entropy solutions Definition
Lebesgue points and the Lebesgue set of an L¹_loc class Definition
Regular distribution from a locally integrable function Definition
Scalar conservation laws, fluxes and Cauchy data Definition
The average of a locally integrable function over a Euclidean ball Definition
Weights, their associated measures, and the spaces Lᵖ(w) Definition
A locally integrable density functional is represented by g dlambda Example
Hilbert transform of the line Poisson kernel Example
Newtonian potential of radial compact data Example
One-dimensional Dirichlet Green kernel on an interval Example
Piecewise-affine approximation of a measurable coefficient Example
BMO functions pair uniformly with H1 atoms Lemma
Bounded compact data give an everywhere finite Newtonian potential Lemma
Dyadic annulus far-field estimates for the maximal function Lemma
Hedberg pointwise inequality for Riesz potentials Lemma
Near and far bounds for a Riesz potential Lemma
Weak solutions factor holomorphically in Hölder coordinates Lemma
Existence and uniqueness for projected dynamics and convergence of projection Euler under countable and dependent choice Theorem
Green representation for classical Poisson data Theorem
Hardy–Littlewood–Sobolev fractional integration inequality Theorem
Lebesgue differentiation theorem on ℝⁿ Theorem
Polynomial growth functions define tempered distributions 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