Locally integrable function

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In mathematics, a locally integrable function (sometimes also called locally summable function)[1] is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.

Definition

Standard definition

Definition 1.[2] Let Ω be an open set in the Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math] and f : Ω → [math]\displaystyle{ \mathbb{C} }[/math] be a Lebesgue measurable function. If f on Ω is such that

[math]\displaystyle{ \int_K | f |\, \mathrm{d}x \lt +\infty, }[/math]

i.e. its Lebesgue integral is finite on all compact subsets K of Ω,[3] then f is called locally integrable. The set of all such functions is denoted by L1,loc(Ω):

[math]\displaystyle{ L_{1,\mathrm{loc}}(\Omega)=\bigl\{f\colon \Omega\to\mathbb{C}\text{ measurable} : f|_K \in L_1(K)\ \forall\, K \subset \Omega,\, K \text{ compact}\bigr\}, }[/math]

where [math]\displaystyle{ \left.f\right|_K }[/math] denotes the restriction of f to the set K.

The classical definition of a locally integrable function involves only measure theoretic and topological[4] concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ):[5] however, since the most common application of such functions is to distribution theory on Euclidean spaces,[2] all the definitions in this and the following sections deal explicitly only with this important case.

An alternative definition

Definition 2.[6] Let Ω be an open set in the Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math]. Then a function f : Ω → [math]\displaystyle{ \mathbb{C} }[/math] such that

[math]\displaystyle{ \int_\Omega | f \varphi|\, \mathrm{d}x \lt +\infty, }[/math]

for each test function φ ∈ C ∞
c (Ω) is called locally integrable, and the set of such functions is denoted by L1,loc(Ω). Here C ∞
c (Ω) denotes the set of all infinitely differentiable functions φ : Ω → [math]\displaystyle{ \mathbb{R} }[/math] 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:[7] it is also the one adopted by (Strichartz 2003) and by (Maz'ya Shaposhnikova).[8] This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:

Lemma 1. A given function f : Ω → [math]\displaystyle{ \mathbb{C} }[/math] is locally integrable according to Definition 1 if and only if it is locally integrable according to Definition 2, i.e.

[math]\displaystyle{ \int_K | f |\, \mathrm{d}x \lt +\infty \quad \forall\, K \subset \Omega,\, K \text{ compact} \quad \Longleftrightarrow \quad \int_\Omega | f \varphi|\, \mathrm{d}x \lt +\infty \quad \forall\, \varphi \in C^\infty_{\mathrm{c}}(\Omega). }[/math]

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

[math]\displaystyle{ \int_\Omega | f \varphi|\, \mathrm{d}x = \int_K |f|\,|\varphi|\, \mathrm{d}x \le\|\varphi\|_\infty\int_K | f |\, \mathrm{d}x\lt \infty }[/math]


Only if part: Let K be a compact subset of the open set Ω. We will first construct a test function φK ∈ C ∞
c (Ω) which majorises the indicator function χK of K. The usual set distance[9] between K and the boundary ∂Ω is strictly greater than zero, i.e.

[math]\displaystyle{ \Delta:=d(K,\partial\Omega)\gt 0, }[/math]

hence it is possible to choose a real number δ such that Δ > 2δ > 0 (if ∂Ω is the empty set, take Δ = ∞). Let Kδ and K2δ denote the closed δ-neighborhood and 2δ-neighborhood of K, respectively. They are likewise compact and satisfy

[math]\displaystyle{ K\subset K_\delta\subset K_{2\delta}\subset\Omega,\qquad d(K_\delta,\partial\Omega)=\Delta-\delta\gt \delta\gt 0. }[/math]

Now use convolution to define the function φK : Ω → [math]\displaystyle{ \mathbb{R} }[/math] by

[math]\displaystyle{ \varphi_K(x)={\chi_{K_\delta}\ast\varphi_\delta(x)}= \int_{\mathbb{R}^n}\chi_{K_\delta}(y)\,\varphi_\delta(x-y)\,\mathrm{d}y, }[/math]

where φδ is a mollifier constructed by using the standard positive symmetric one. Obviously φK is non-negative in the sense that φK ≥ 0, infinitely differentiable, and its support is contained in K2δ, in particular it is a test function. Since φK(x) = 1 for all x ∈ K, we have that χK ≤ φK.

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

[math]\displaystyle{ \int_K|f|\,\mathrm{d}x=\int_\Omega|f|\chi_K\,\mathrm{d}x \le\int_\Omega|f|\varphi_K\,\mathrm{d}x\lt \infty. }[/math]

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

Definition 3.[10] Let Ω be an open set in the Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math] and f : Ω → [math]\displaystyle{ \mathbb{C} }[/math] be a Lebesgue measurable function. If, for a given p with 1 ≤ p ≤ +∞, f satisfies

[math]\displaystyle{ \int_K | f|^p \,\mathrm{d}x \lt +\infty, }[/math]

i.e., it belongs to Lp(K) for all compact subsets K of Ω, then f is called locally p-integrable or also p-locally integrable.[10] The set of all such functions is denoted by Lp,loc(Ω):

[math]\displaystyle{ L_{p,\mathrm{loc}}(\Omega)=\left\{f:\Omega\to\mathbb{C}\text{ measurable }\left|\ f|_K \in L_p(K),\ \forall\, K \subset \Omega, K \text{ compact}\right.\right\}. }[/math]

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.[11] Despite their apparent higher generality, locally p-integrable functions form a subset of locally integrable functions for every p such that 1 < p ≤ +∞.[12]

Notation

Apart from the different glyphs which may be used for the uppercase "L",[13] there are few variants for the notation of the set of locally integrable functions

  • [math]\displaystyle{ L^p_{\mathrm{loc}}(\Omega), }[/math] adopted by (Hörmander 1990), (Strichartz 2003) and (Vladimirov 2002).
  • [math]\displaystyle{ L_{p,\mathrm{loc}}(\Omega), }[/math] adopted by (Maz'ya Poborchi) and (Maz'ya Shaposhnikova).
  • [math]\displaystyle{ L_p(\Omega,\mathrm{loc}), }[/math] adopted by (Maz'ja 1985) and (Maz'ya 2011).

Properties

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

Theorem 1.[14] Lp,loc is a complete metrizable space: its topology can be generated by the following metric:

[math]\displaystyle{ d(u,v)=\sum_{k\geq 1}\frac{1}{2^k}\frac{\Vert u - v\Vert_{p,\omega_k}}{1+\Vert u - v\Vert_{p,\omega_k}}\qquad u, v\in L_{p,\mathrm{loc}}(\Omega), }[/math]

where {ωk}k≥1 is a family of non empty open sets such that

  • ωk ⊂⊂ ωk+1, meaning that ωk is compactly included in ωk+1 i.e. it is a set having compact closure strictly included in the set of higher index.
  • ∪kωk = Ω.
  • [math]\displaystyle{ \scriptstyle{\Vert\cdot\Vert_{p,\omega_k}}\to\mathbb{R}^+ }[/math], k ∈ [math]\displaystyle{ \mathbb{N} }[/math] is an indexed family of seminorms, defined as
[math]\displaystyle{ {\Vert u \Vert_{p,\omega_k}} = \left (\int_{\omega_k} | u(x)|^p \,\mathrm{d}x\right)^{1/p}\qquad\forall\, u\in L_{p,\mathrm{loc}}(\Omega). }[/math]

In references (Gilbarg Trudinger), (Maz'ya Poborchi), (Maz'ja 1985) and (Maz'ya 2011), this theorem is stated but not proved on a formal basis:[15] a complete proof of a more general result, which includes it, is found in (Meise Vogt).

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

Theorem 2. Every function f belonging to Lp(Ω), 1 ≤ p ≤ +∞, where Ω is an open subset of [math]\displaystyle{ \mathbb{R}^n }[/math], 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 χK of a compact subset K of Ω: then, for p ≤ +∞,

[math]\displaystyle{ \left|{\int_\Omega|\chi_K|^q\,\mathrm{d}x}\right|^{1/q}=\left|{\int_K \mathrm{d}x}\right|^{1/q}=|K|^{1/q}\lt +\infty, }[/math]

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 Lp(Ω), by Hölder's inequality, the product fχK is integrable i.e. belongs to L1(Ω) and

[math]\displaystyle{ {\int_K|f|\,\mathrm{d}x}={\int_\Omega|f\chi_K|\,\mathrm{d}x}\leq\left|{\int_\Omega|f|^p\,\mathrm{d}x}\right|^{1/p}\left|{\int_K \mathrm{d}x}\right|^{1/q}=\|f\|_p|K|^{1/q}\lt +\infty, }[/math]

therefore

[math]\displaystyle{ f\in L_{1,\mathrm{loc}}(\Omega). }[/math]

Note that since the following inequality is true

[math]\displaystyle{ {\int_K|f|\,\mathrm{d}x}={\int_\Omega|f\chi_K|\,\mathrm{d}x}\leq\left|{\int_K|f|^p \,\mathrm{d}x}\right|^{1/p}\left|{\int_K \mathrm{d}x}\right|^{1/q}=\|f \chi_K\|_p|K|^{1/q}\lt +\infty, }[/math]

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.

Corollary 1. Every function [math]\displaystyle{ f }[/math] in [math]\displaystyle{ L_{p,loc}(\Omega) }[/math], [math]\displaystyle{ 1\lt p\leq\infty }[/math], is locally integrable, i. e. belongs to [math]\displaystyle{ L_{1,loc}(\Omega) }[/math].

Note: If [math]\displaystyle{ \Omega }[/math] is an open subset of [math]\displaystyle{ \mathbb{R}^n }[/math] that is also bounded, then one has the standard inclusion [math]\displaystyle{ L_p(\Omega) \subset L_1(\Omega) }[/math] which makes sense given the above inclusion [math]\displaystyle{ L_1(\Omega)\subset L_{1,loc}(\Omega) }[/math]. But the first of these statements is not true if [math]\displaystyle{ \Omega }[/math] is not bounded; then it is still true that [math]\displaystyle{ L_p(\Omega) \subset L_{1,loc}(\Omega) }[/math] for any [math]\displaystyle{ p }[/math], but not that [math]\displaystyle{ L_p(\Omega)\subset L_1(\Omega) }[/math]. To see this, one typically considers the function [math]\displaystyle{ u(x)=1 }[/math], which is in [math]\displaystyle{ L_{\infty}(\mathbb{R}^n) }[/math] but not in [math]\displaystyle{ L_p(\mathbb{R}^n) }[/math] for any finite [math]\displaystyle{ p }[/math].

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

Theorem 3. A function f is the density of an absolutely continuous measure if and only if [math]\displaystyle{ f\in L_{1,loc} }[/math].

The proof of this result is sketched by (Schwartz 1998). 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.[16]

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[17] and integrable functions are locally integrable.[18]
  • The function [math]\displaystyle{ f(x) = 1/x }[/math] 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
[math]\displaystyle{ f(x)= \begin{cases} 1/x &x\neq 0,\\ 0 & x=0, \end{cases} \quad x \in \mathbb R }[/math]
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, [math]\displaystyle{ 1/x \in L_{1, loc}(\mathbb{R}\setminus 0) }[/math]:[19] however, this function can be extended to a distribution on the whole [math]\displaystyle{ \mathbb{R} }[/math] as a Cauchy principal value.[20]
  • The preceding example raises a question: does every function which is locally integrable in Ω ⊊ [math]\displaystyle{ \mathbb{R} }[/math] admit an extension to the whole [math]\displaystyle{ \mathbb{R} }[/math] as a distribution? The answer is negative, and a counterexample is provided by the following function:
[math]\displaystyle{ f(x)= \begin{cases} e^{1/x} &x\neq 0,\\ 0 & x=0, \end{cases} }[/math]
does not define any distribution on [math]\displaystyle{ \mathbb{R} }[/math].[21]
  • The following example, similar to the preceding one, is a function belonging to L1,loc([math]\displaystyle{ \mathbb{R} }[/math] \ 0) which serves as an elementary counterexample in the application of the theory of distributions to differential operators with irregular singular coefficients:
[math]\displaystyle{ f(x)= \begin{cases} k_1 e^{1/x^2} &x\gt 0,\\ 0 & x=0,\\ k_2 e^{1/x^2} &x\lt 0, \end{cases} }[/math]
where k1 and k2 are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order
[math]\displaystyle{ x^3\frac{\mathrm{d}f}{\mathrm{d}x}+2f=0. }[/math]
Again it does not define any distribution on the whole [math]\displaystyle{ \mathbb{R} }[/math], 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.[22]

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.

See also

Notes

  1. ↑ According to (Gel'fand Shilov).
  2. ↑ 2.0 2.1 See for example (Schwartz 1998) and (Vladimirov 2002).
  3. ↑ Another slight variant of this definition, chosen by (Vladimirov 2002), is to require only that K ⋐ Ω (or, using the notation of (Gilbarg Trudinger), K ⊂⊂ Ω), meaning that K is strictly included in Ω i.e. it is a set having compact closure strictly included in the given ambient set.
  4. ↑ The notion of compactness must obviously be defined on the given abstract measure space.
  5. ↑ This is the approach developed for example by (Cafiero 1959) and by (Saks 1937), without dealing explicitly with the locally integrable case.
  6. ↑ See for example (Strichartz 2003).
  7. ↑ This approach was praised by (Schwartz 1998) who remarked also its usefulness, however using Definition 1 to define locally integrable functions.
  8. ↑ Be noted that Maz'ya and Shaposhnikova define explicitly only the "localized" version of the Sobolev space Wk,p(Ω), nevertheless explicitly asserting that the same method is used to define localized versions of all other Banach spaces used in the cited book: in particular, Lp,loc(Ω) is introduced on page 44.
  9. ↑ Not to be confused with the Hausdorff distance.
  10. ↑ 10.0 10.1 See for example (Vladimirov 2002) and (Maz'ya Poborchi).
  11. ↑ As remarked in the previous section, this is the approach adopted by (Maz'ya Shaposhnikova), without developing the elementary details.
  12. ↑ Precisely, they form a vector subspace of L1,loc(Ω): see Corollary 1 to Theorem 2.
  13. ↑ See for example (Vladimirov 2002), where a calligraphic ℒ is used.
  14. ↑ See (Gilbarg Trudinger), (Maz'ya Poborchi) for a statement of this results, and also the brief notes in (Maz'ja 1985) and (Maz'ya 2011).
  15. ↑ (Gilbarg Trudinger) and (Maz'ya Poborchi) only sketch very briefly the method of proof, while in (Maz'ja 1985) and (Maz'ya 2011) it is assumed as a known result, from which the subsequent development starts.
  16. ↑ According to (Saks 1937), "If E is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure (μ), then, in order that an additive function of a set (𝔛) on E be absolutely continuous on E, it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of E". Assuming (μ) to be the Lebesgue measure, the two statements can be seen to be equivalent.
  17. ↑ See for example (Hörmander 1990).
  18. ↑ See (Strichartz 2003).
  19. ↑ See (Schwartz 1998).
  20. ↑ See (Vladimirov 2002).
  21. ↑ See (Vladimirov 2002).
  22. ↑ For a brief discussion of this example, see (Schwartz 1998).

References

External links



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