Locally integrable function

Definition

Standard definition

. Let Ω be an open set in the Euclidean space $R^{n}$ and f : Ω → $C$ be a Lebesgue measurable function. If f on Ω is such that

\int_{K} | f | d x < + \infty,

i.e. its Lebesgue integral is finite on all compact subsets K of Ω, then f is called locally integrable. The set of all such functions is denoted by L_1,loc(Ω):

L_{1, l o c} (Ω) = {f : Ω \to C measurable : f |_{K} \in L_{1} (K) \forall K \subset Ω, K compact},

where ${f |}_{K}$ denotes the restriction of f to the set K.

The classical definition of a locally integrable function involves only measure theoretic and topological concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ): however, since the most common application of such functions is to distribution theory on Euclidean spaces, all the definitions in this and the following sections deal explicitly only with this important case.

An alternative definition

. Let Ω be an open set in the Euclidean space $R^{n}$ . Then a function f : Ω → $C$ such that

\int_{Ω} | f φ | d x < + \infty,

for each test function φ ∈ C ∞
c (Ω) is called locally integrable, and the set of such functions is denoted by L_1,loc(Ω). Here C ∞
c (Ω) denotes the set of all infinitely differentiable functions φ : Ω → $R$ with compact support contained in Ω.

This definition has its roots in the approach to measure and integration theory based on the concept of continuous linear functional on a topological vector space, developed by the Nicolas Bourbaki school: it is also the one adopted by Strichartz (2003) and by Maz'ya & Shaposhnikova (2009, p. 34). This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:

. A given function f : Ω → $C$ is locally integrable according to Definition 1 if and only if it is locally integrable according to Definition 2, i.e.

\int_{K} | f | d x < + \infty \forall K \subset Ω, K compact ⟺ \int_{Ω} | f φ | d x < + \infty \forall φ \in C_{c}^{\infty} (Ω) .

Proof of Lemma 1

If part: Let φ ∈ C ∞
c (Ω) be a test function. It is bounded by its supremum norm ||φ||_∞, measurable, and has a compact support, let's call it K. Hence

\int_{Ω} | f φ | d x = \int_{K} | f | | φ | d x \leq ‖ φ ‖_{\infty} \int_{K} | f | d x < \infty

by Definition 1.

Only if part: Let K be a compact subset of the open set Ω. We will first construct a test function φ_K ∈ C ∞
c (Ω) which majorises the indicator function χ_K of K. The usual set distance between K and the boundary ∂Ω is strictly greater than zero, i.e.

Δ := d (K, \partial Ω) > 0,

hence it is possible to choose a real number δ such that Δ > 2δ > 0 (if ∂Ω is the empty set, take Δ = ∞). Let K_δ and K_2δ denote the closed δ-neighborhood and 2δ-neighborhood of K, respectively. They are likewise compact and satisfy

K \subset K_{δ} \subset K_{2 δ} \subset Ω, d (K_{δ}, \partial Ω) = Δ - δ > δ > 0.

Now use convolution to define the function φ_K : Ω → $R$ by

φ_{K} (x) = χ_{K_{δ}} * φ_{δ} (x) = \int_{R^{n}} χ_{K_{δ}} (y) φ_{δ} (x - y) d y,

where φ_δ is a mollifier constructed by using the standard positive symmetric one. Obviously φ_K is non-negative in the sense that φ_K ≥ 0, infinitely differentiable, and its support is contained in K_2δ, in particular it is a test function. Since φ_K(x) = 1 for all x ∈ K, we have that χ_K ≤ φ_K.

Let f be a locally integrable function according to Definition 2. Then

\int_{K} | f | d x = \int_{Ω} | f | χ_{K} d x \leq \int_{Ω} | f | φ_{K} d x < \infty .

Since this holds for every compact subset K of Ω, the function f is locally integrable according to Definition 1. □

Generalization: locally p-integrable functions

. Let Ω be an open set in the Euclidean space $R^{n}$ and f : Ω → $C$ be a Lebesgue measurable function. If, for a given p with 1 ≤ p ≤ +∞, f satisfies

\int_{K} | f |^{p} d x < + \infty,

i.e., it belongs to L_p(K) for all compact subsets K of Ω, then f is called locally p-integrable or also p-locally integrable. The set of all such functions is denoted by L_p,loc(Ω):

L_{p, l o c} (Ω) = {f : Ω \to C measurable | f |_{K} \in L_{p} (K), \forall K \subset Ω, K compact} .

An alternative definition, completely analogous to the one given for locally integrable functions, can also be given for locally p-integrable functions: it can also be and proven equivalent to the one in this section. Despite their apparent higher generality, locally p-integrable functions form a subset of locally integrable functions for every p such that 1 < p ≤ +∞.

Notation

Apart from the different glyphs which may be used for the uppercase "L", there are few variants for the notation of the set of locally integrable functions

$L_{l o c}^{p} (Ω),$ adopted by (Hörmander 1990, p. 37), (Strichartz 2003, pp. 12–13) and (Vladimirov 2002, p. 3).
$L_{p, l o c} (Ω),$ adopted by (Maz'ya & Poborchi 1997, p. 4) and Maz'ya & Shaposhnikova (2009, p. 44).
$L_{p} (Ω, l o c),$ adopted by (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2).

3 more chapters are free with a Lably account.

Properties

Lp,loc is a complete metric space for all p ≥ 1

. L_p,loc is a complete metrizable space: its topology can be generated by the following metric:

d (u, v) = \sum_{k \geq 1} \frac{1}{2^{k}} \frac{‖ u - v ‖_{p, ω_{k}}}{1 + ‖ u - v ‖_{p, ω_{k}}} u, v \in L_{p, l o c} (Ω),

where {ω_k}_k≥1 is a family of non empty open sets such that

ω_k ⊂⊂ ω_k+1, meaning that ω_k is compactly included in ω_k+1 i.e. it is a set having compact closure strictly included in the set of higher index.
∪_kω_k = Ω.
$‖ \cdot ‖_{p, ω_{k}} \to R^{+}$ , k ∈ $N$ is an indexed family of seminorms, defined as

‖ u ‖_{p, ω_{k}} = {(\int_{ω_{k}} | u (x) |^{p} d x)}^{1 / p} \forall u \in L_{p, l o c} (Ω) .

In references (Gilbarg & Trudinger 2001, p. 147), (Maz'ya & Poborchi 1997, p. 5), (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2), this theorem is stated but not proved on a formal basis: a complete proof of a more general result, which includes it, is found in (Meise & Vogt 1997, p. 40).

Lp is a subspace of L1,loc for all p ≥ 1

. Every function f belonging to L_p(Ω), 1 ≤ p ≤ +∞, where Ω is an open subset of $R^{n}$ , is locally integrable.

Proof. The case p = 1 is trivial, therefore in the sequel of the proof it is assumed that 1 < p ≤ +∞. Consider the characteristic function χ_K of a compact subset K of Ω: then, for p ≤ +∞,

{| \int_{Ω} | χ_{K} |^{q} d x |}^{1 / q} = {| \int_{K} d x |}^{1 / q} = | K |^{1 / q} < + \infty,

where

q is a positive number such that 1/p + 1/q = 1 for a given 1 ≤ p ≤ +∞
|K| is the Lebesgue measure of the compact set K

Then for any f belonging to L_p(Ω), by Hölder's inequality, the product fχ_K is integrable i.e. belongs to L₁(Ω) and

\int_{K} | f | d x = \int_{Ω} | f χ_{K} | d x \leq {| \int_{Ω} | f |^{p} d x |}^{1 / p} {| \int_{K} d x |}^{1 / q} = ‖ f ‖_{p} | K |^{1 / q} < + \infty,

therefore

f \in L_{1, l o c} (Ω) .

Note that since the following inequality is true

\int_{K} | f | d x = \int_{Ω} | f χ_{K} | d x \leq {| \int_{K} | f |^{p} d x |}^{1 / p} {| \int_{K} d x |}^{1 / q} = ‖ f χ_{K} ‖_{p} | K |^{1 / q} < + \infty,

the theorem is true also for functions f belonging only to the space of locally p-integrable functions, therefore the theorem implies also the following result.

. Every function $f$ in $L_{p, l o c} (Ω)$ , $1 < p \leq \infty$ , is locally integrable, i. e. belongs to $L_{1, l o c} (Ω)$ .

Note: If $Ω$ is an open subset of $R^{n}$ that is also bounded, then one has the standard inclusion $L_{p} (Ω) \subset L_{1} (Ω)$ which makes sense given the above inclusion $L_{1} (Ω) \subset L_{1, l o c} (Ω)$ . But the first of these statements is not true if $Ω$ is not bounded; then it is still true that $L_{p} (Ω) \subset L_{1, l o c} (Ω)$ for any $p$ , but not that $L_{p} (Ω) \subset L_{1} (Ω)$ . To see this, one typically considers the function $u (x) = 1$ , which is in $L_{\infty} (R^{n})$ but not in $L_{p} (R^{n})$ for any finite $p$ .

L1,loc is the space of densities of absolutely continuous measures

. A function f is the density of an absolutely continuous measure if and only if $f \in L_{1, l o c}$ .

The proof of this result is sketched by (Schwartz 1998, p. 18). Rephrasing its statement, this theorem asserts that every locally integrable function defines an absolutely continuous measure and conversely that every absolutely continuous measures defines a locally integrable function: this is also, in the abstract measure theory framework, the form of the important Radon–Nikodym theorem given by Stanisław Saks in his treatise.

Examples

The constant function 1 defined on the real line is locally integrable but not globally integrable since the real line has infinite measure. More generally, constants, continuous functions and integrable functions are locally integrable.
The function $f (x) = 1 / x$ for x ∈ (0, 1) is locally but not globally integrable on (0, 1). It is locally integrable since any compact set K ⊆ (0, 1) has positive distance from 0 and f is hence bounded on K. This example underpins the initial claim that locally integrable functions do not require the satisfaction of growth conditions near the boundary in bounded domains.
The function

f (x) = {\begin{cases} 1 / x & x \neq 0, \\ 0 & x = 0, \end{cases} x \in R

is not locally integrable in x = 0: it is indeed locally integrable near this point since its integral over every compact set not including it is finite. Formally speaking,

1 / x \in L_{1, l o c} (R ∖ 0)

: however, this function can be extended to a distribution on the whole

R

as a Cauchy principal value.

The preceding example raises a question: does every function which is locally integrable in Ω ⊊ $R$ admit an extension to the whole $R$ as a distribution? The answer is negative, and a counterexample is provided by the following function:

f (x) = {\begin{cases} e^{1 / x} & x \neq 0, \\ 0 & x = 0, \end{cases}

does not define any distribution on

R

The following example, similar to the preceding one, is a function belonging to L_1,loc( $R$ \ 0) which serves as an elementary counterexample in the application of the theory of distributions to differential operators with irregular singular coefficients:

f (x) = {\begin{cases} k_{1} e^{1 / x^{2}} & x > 0, \\ 0 & x = 0, \\ k_{2} e^{1 / x^{2}} & x < 0, \end{cases}

where k₁ and k₂ are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order

x^{3} \frac{d f}{d x} + 2 f = 0.

Again it does not define any distribution on the whole

R

, if k₁ or k₂ are not zero: the only distributional global solution of such equation is therefore the zero distribution, and this shows how, in this branch of the theory of differential equations, the methods of the theory of distributions cannot be expected to have the same success achieved in other branches of the same theory, notably in the theory of linear differential equations with constant coefficients.

Applications

Locally integrable functions play a prominent role in distribution theory and they occur in the definition of various classes of functions and function spaces, like functions of bounded variation. Moreover, they appear in the Radon–Nikodym theorem by characterizing the absolutely continuous part of every measure.