Locally integrable function

Template:Short description In mathematics, a locally integrable function (sometimes also called locally summable function) 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 space|Template:Math 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

Template:EquationRef. Let Template:Math be an open set in the Euclidean space ${ℝ}^n$ and Template:Math be a Lebesgue measurable function. If Template:Math on Template:Math is such that

$\underset{K}{\int }|f|\mathrm{d}x<+\mathrm{\infty },$

i.e. its Lebesgue integral is finite on all compact subsets Template:Math of Template:Math, then Template:Math is called locally integrable. The set of all such functions is denoted by Template:Math:

$L_{1,\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })=\{f:\mathrm{\Omega }\to ℂ\text{ measurable}:f{|}_K\in L_1(K)\mathrm{\forall }K\subset \mathrm{\Omega },K\text{ compact}\},$

where ${f|}_K$ denotes the restriction of Template:Math to the set Template:Math.

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 Template:Math: 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

Template:EquationRef. Let Template:Math be an open set in the Euclidean space ${ℝ}^n$. Then a function Template:Math such that

$\underset{\mathrm{\Omega }}{\int }|f\phi |\mathrm{d}x<+\mathrm{\infty },$

for each test function Template:Math is called locally integrable, and the set of such functions is denoted by Template:Math. Here Template:Math denotes the set of all infinitely differentiable functions Template:Math with compact support contained in Template:Math.

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 Template:Harvtxt and by Template:Harvtxt. This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:

Template:EquationRef. A given function Template:Math is locally integrable according to Template:EquationNote if and only if it is locally integrable according to Template:EquationNote, i.e.

$\underset{K}{\int }|f|\mathrm{d}x<+\mathrm{\infty }\mathrm{\forall }K\subset \mathrm{\Omega },K\text{ compact}⟺\underset{\mathrm{\Omega }}{\int }|f\phi |\mathrm{d}x<+\mathrm{\infty }\mathrm{\forall }\phi \in C_{\mathrm{c}}^{\mathrm{\infty }}(\mathrm{\Omega }).$

Proof of Template:EquationNote

If part: Let Template:Math be a test function. It is bounded by its supremum norm Template:Math, measurable, and has a compact support, let's call it Template:Math. Hence

$\underset{\mathrm{\Omega }}{\int }|f\phi |\mathrm{d}x=\underset{K}{\int }|f||\phi |\mathrm{d}x\le \Vert \phi {\Vert }_{\mathrm{\infty }}\underset{K}{\int }|f|\mathrm{d}x<\mathrm{\infty }$

by Template:EquationNote.

Only if part: Let Template:Math be a compact subset of the open set Template:Math. We will first construct a test function Template:Math which majorises the indicator function Template:Math of Template:Math. The usual set distance between Template:Math and the boundary Template:Math is strictly greater than zero, i.e.

$\mathrm{\Delta }:=d(K,\partial \mathrm{\Omega })>0,$

hence it is possible to choose a real number Template:Math such that Template:Math (if Template:Math is the empty set, take Template:Math). Let Template:Math and Template:Math denote the closed [[Neighbourhood (mathematics)#In a metric space|Template:Math-neighborhood]] and Template:Math-neighborhood of Template:Math, respectively. They are likewise compact and satisfy

$K\subset K_{\delta }\subset K_{2\delta }\subset \mathrm{\Omega },d(K_{\delta },\partial \mathrm{\Omega })=\mathrm{\Delta }-\delta >\delta >0.$

Now use convolution to define the function Template:Math by

${\phi }_K(x)={\chi }_{K_{\delta }}\ast {\phi }_{\delta }(x)=\underset{{ℝ}^n}{\int }{\chi }_{K_{\delta }}(y){\phi }_{\delta }(x-y)\mathrm{d}y,$

where Template:Math is a mollifier constructed by using the standard positive symmetric one. Obviously Template:Math is non-negative in the sense that Template:Math, infinitely differentiable, and its support is contained in Template:Math, in particular it is a test function. Since Template:Math for all Template:Math, we have that Template:Math.

Let Template:Math be a locally integrable function according to Template:EquationNote. Then

$\underset{K}{\int }|f|\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }|f|{\chi }_K\mathrm{d}x\le \underset{\mathrm{\Omega }}{\int }|f|{\phi }_K\mathrm{d}x<\mathrm{\infty }.$

Since this holds for every compact subset Template:Math of Template:Math, the function Template:Math is locally integrable according to Template:EquationNote. □

Generalization: locally p-integrable functions

Template:EquationRef. Let Template:Math be an open set in the Euclidean space ${ℝ}^n$ and Template:Math$ℂ$ be a Lebesgue measurable function. If, for a given Template:Math with Template:Math, Template:Math satisfies

$\underset{K}{\int }|f{|}^p\mathrm{d}x<+\mathrm{\infty },$

i.e., it belongs to Template:Math for all compact subsets Template:Math of Template:Math, then Template:Math is called locally Template:Math-integrable or also Template:Math-locally integrable. The set of all such functions is denoted by Template:Math:

$L_{p,\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })=\{f:\mathrm{\Omega }\to ℂ\text{ measurable }|f{|}_K\in L_p(K),\mathrm{\forall }K\subset \mathrm{\Omega },K\text{ compact}\}.$

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

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

• $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p(\mathrm{\Omega }),$ adopted by Template:Harv, Template:Harv and Template:Harv.
• $L_{p,\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega }),$ adopted by Template:Harv and Template:Harvtxt.
• $L_p(\mathrm{\Omega },\mathrm{l}\mathrm{o}\mathrm{c}),$ adopted by Template:Harv and Template:Harv.

Properties

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

Template:EquationRef. Template:Math is a complete metrizable space: its topology can be generated by the following metric:

$d(u,v)=\sum _{k\ge 1}\frac{1}{2^k}\frac{\Vert u-v{\Vert }_{p,{\omega }_k}}{1+\Vert u-v{\Vert }_{p,{\omega }_k}}u,v\in L_{p,\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega }),$

where Template:Math is a family of non empty open sets such that

• Template:Math, meaning that Template:Math is compactly included in Template:Math i.e. it is a set having compact closure strictly included in the set of higher index.
• Template:Math.
• $\Vert \cdot {\Vert }_{p,{\omega }_k}\to {ℝ}^{+}$, k ∈ $\mathbb{N}$ is an indexed family of seminorms, defined as

$\Vert u{\Vert }_{p,{\omega }_k}={(\underset{{\omega }_k}{\int }|u(x){|}^p\mathrm{d}x)}^{1/p}\mathrm{\forall }u\in L_{p,\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega }).$

In references Template:Harv, Template:Harv, Template:Harv and Template:Harv, 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 Template:Harv.

L_p is a subspace of L_1,loc for all p ≥ 1

Template:EquationRef. Every function Template:Math belonging to Template:Math, Template:Math, where Template:Math is an open subset of ${ℝ}^n$, is locally integrable.

Proof. The case Template:Math is trivial, therefore in the sequel of the proof it is assumed that Template:Math. Consider the characteristic function Template:Math of a compact subset Template:Math of Template:Math: then, for Template:Math,

${\left|\underset{\mathrm{\Omega }}{\int }|{\chi }_K{|}^q\mathrm{d}x\right|}^{1/q}={\left|\underset{K}{\int }\mathrm{d}x\right|}^{1/q}=|K{|}^{1/q}<+\mathrm{\infty },$

where

• Template:Math is a positive number such that Template:Math = Template:Math for a given Template:Math
• Template:Math is the Lebesgue measure of the compact set Template:Math

Then for any Template:Math belonging to Template:Math, by Hölder's inequality, the product Template:Math is integrable i.e. belongs to Template:Math and

$\underset{K}{\int }|f|\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }|f{\chi }_K|\mathrm{d}x\le {\left|\underset{\mathrm{\Omega }}{\int }|f{|}^p\mathrm{d}x\right|}^{1/p}{\left|\underset{K}{\int }\mathrm{d}x\right|}^{1/q}=\Vert f{\Vert }_p|K{|}^{1/q}<+\mathrm{\infty },$

therefore

$f\in L_{1,\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega }).$

Note that since the following inequality is true

$\underset{K}{\int }|f|\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }|f{\chi }_K|\mathrm{d}x\le {\left|\underset{K}{\int }|f{|}^p\mathrm{d}x\right|}^{1/p}{\left|\underset{K}{\int }\mathrm{d}x\right|}^{1/q}=\Vert f{\chi }_K{\Vert }_p|K{|}^{1/q}<+\mathrm{\infty },$

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

Template:EquationRef. Every function $f$ in $L_{p,loc}(\mathrm{\Omega })$, $1<p\le \mathrm{\infty }$, is locally integrable, i. e. belongs to $L_{1,loc}(\mathrm{\Omega })$.

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

L_1,loc is the space of densities of absolutely continuous measures

Template:EquationRef. A function Template:Math is the density of an absolutely continuous measure if and only if $f\in L_{1,loc}$.

The proof of this result is sketched by Template:Harv. 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 Template:Math 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

$f(x)=\{\begin{array}{cc}1/x& x\ne 0,\\ 0& x=0,\end{array}x\in ℝ$

is not locally integrable in Template:Math: it is indeed locally integrable near this point since its integral over every compact set not including it is finite. Formally speaking, $1/x\in L_{1,loc}(ℝ\setminus 0)$: 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 Template:Math ⊊ $ℝ$ admit an extension to the whole $ℝ$ as a distribution? The answer is negative, and a counterexample is provided by the following function:

$f(x)=\{\begin{array}{cc}{e}^{1/x}& x\ne 0,\\ 0& x=0,\end{array}$

does not define any distribution on $ℝ$.

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

$f(x)=\{\begin{array}{cc}{k}_1{e}^{1/x^2}& x>0,\\ 0& x=0,\\ k_2{e}^{1/x^2}& x<0,\end{array}$

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

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

Again it does not define any distribution on the whole $ℝ$, if Template:Math or Template:Math 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.

See also

• Compact set
• Distribution (mathematics)
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Lebesgue integral
• Lp space

References

• Template:Citation. Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
• Template:Citation. Translated from the original 1958 Russian edition by Eugene Saletan, this is an important monograph on the theory of generalized functions, dealing both with distributions and analytic functionals.
• Template:Citation.
• Template:Citation (available also as Template:ISBN).
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• Template:Citation. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
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• Template:Citation. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.

External links

• Template:MathWorld
• Template:Springer