locally integrable function

{{EquationRef|1|Definition 1}}.See for example {{Harv|Schwartz|1998|p=18}} and {{Harv|Vladimirov|2002|p=3}}. Let {{math|Ω}} be an open set in the Euclidean space $\backslash mathbb\{R\}^n$ and {{math|f : Ω → $\backslash mathbb\{C\}$}} be a Lebesgue measurable function. If {{math|f}} on {{math|Ω}} is such that

:

i.e. its Lebesgue integral is finite on all compact subsets {{math|K}} of {{math|Ω}},Another slight variant of this definition, chosen by {{harvtxt|Vladimirov|2002|p=1}}, is to require only that {{math|K ⋐ Ω}} (or, using the notation of {{harvtxt|Gilbarg|Trudinger|2001|p=9}}, {{math|K ⊂⊂ Ω}}), meaning that {{math|K}} is strictly included in {{math|Ω}} i.e. it is a set having compact closure strictly included in the given ambient set. then {{math|f}} is called locally integrable. The set of all such functions is denoted by {{math|L_1,loc(Ω)}}:

:$L\_\{1,\backslash mathrm\{loc\}\}(\backslash Omega)=\backslash bigl\backslash \{f\backslash colon\; \backslash Omega\backslash to\backslash mathbb\{C\}\backslash text\{\; measurable\}\; :\; f|\_K\; \backslash in\; L\_1(K)\backslash \; \backslash forall\backslash ,\; K\; \backslash subset\; \backslash Omega,\backslash ,\; K\; \backslash text\{\; compact\}\backslash bigr\backslash \},$

where $\backslash left.f\backslash right|\_K$ denotes the restriction of {{math|f}} to the set {{math|K}}.

The classical definition of a locally integrable function involves only measure theoretic and topologicalThe notion of compactness must obviously be defined on the given abstract measure space. concepts and can be carried over abstract to complex-valued functions on a topological measure space {{math|(X, Σ, μ)}}:This is the approach developed for example by {{harvtxt|Cafiero|1959|pp=285–342}} and by {{harvtxt|Saks|1937|loc = chapter I}}, without dealing explicitly with the locally integrable case. 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.

{{EquationRef|2|Definition 2}}.See for example {{Harv|Strichartz|2003|pp=12–13}}. Let {{math|Ω}} be an open set in the Euclidean space $\backslash mathbb\{R\}^n$. Then a function {{math|f : Ω → $\backslash mathbb\{C\}$}} such that

:

for each test function {{math|φ ∈ {{SubSup|C|c|∞}}(Ω)}} is called locally integrable, and the set of such functions is denoted by {{math|L_1,loc(Ω)}}. Here {{math|{{SubSup|C|c|∞}}(Ω)}} denotes the set of all infinitely differentiable functions {{math|φ : Ω → $\backslash mathbb\{R\}$}} with compact support contained in {{math|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:This approach was praised by {{harvtxt|Schwartz|1998|pp=16–17}} who remarked also its usefulness, however using {{EquationNote|1|Definition 1}} to define locally integrable functions. it is also the one adopted by {{Harvtxt|Strichartz|2003}} and by {{Harvtxt|Maz'ya|Shaposhnikova|2009|p=34}}.Be noted that Maz'ya and Shaposhnikova define explicitly only the "localized" version of the Sobolev space {{math|W^k,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, {{math|L_p,loc(Ω)}} is introduced on page 44. This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:

{{EquationRef|3|Lemma 1}}. A given function {{math|f : Ω → $\backslash mathbb\{C\}$}} is locally integrable according to {{EquationNote|1|Definition 1}} if and only if it is locally integrable according to {{EquationNote|2|Definition 2}}, i.e.

:

\int_\Omega | f \varphi|\, \mathrm{d}x <+\infty \quad \forall\, \varphi \in C^\infty_{\mathrm{c}}(\Omega).

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

:

by {{EquationNote|1|Definition 1}}.

Only if part: Let {{math|K}} be a compact subset of the open set {{math|Ω}}. We will first construct a test function {{math|φ_K ∈ {{SubSup|C|c|∞}}(Ω)}} which majorises the indicator function {{math|χ_K}} of {{math|K}}.

The usual set distanceNot to be confused with the Hausdorff distance. between {{math|K}} and the boundary {{math|∂Ω}} is strictly greater than zero, i.e.

:$\backslash Delta:=d(K,\backslash partial\backslash Omega)>0,$

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

:$K\backslash subset\; K\_\backslash delta\backslash subset\; K\_\{2\backslash delta\}\backslash subset\backslash Omega,\backslash qquad\; d(K\_\backslash delta,\backslash partial\backslash Omega)=\backslash Delta-\backslash delta>\backslash delta>0.$

Now use convolution to define the function {{math|φ_K : Ω → $\backslash mathbb\{R\}$}} by

:$\backslash varphi\_K(x)=\{\backslash chi\_\{K\_\backslash delta\}\backslash ast\backslash varphi\_\backslash delta(x)\}=$

\int_{\mathbb{R}^n}\chi_{K_\delta}(y)\,\varphi_\delta(x-y)\,\mathrm{d}y,

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

Let {{math|f}} be a locally integrable function according to {{EquationNote|2|Definition 2}}. Then

:$\backslash int\_K|f|\backslash ,\backslash mathrm\{d\}x=\backslash int\_\backslash Omega|f|\backslash chi\_K\backslash ,\backslash mathrm\{d\}x$

\le\int_\Omega|f|\varphi_K\,\mathrm{d}x<\infty.

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

{{EquationRef|4|Definition 3}}.See for example {{Harv|Vladimirov|2002|p=3}} and {{harv|Maz'ya|Poborchi|1997|p=4}}. Let {{math|Ω}} be an open set in the Euclidean space $\backslash mathbb\{R\}^n$ and {{math|f : Ω → }}$\backslash mathbb\{C\}$ be a Lebesgue measurable function. If, for a given {{math|p}} with {{math|1 ≤ p ≤ +∞}}, {{math|f}} satisfies

:

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

:$L\_\{p,\backslash mathrm\{loc\}\}(\backslash Omega)=\backslash left\backslash \{f:\backslash Omega\backslash to\backslash mathbb\{C\}\backslash text\{\; measurable\; \}\backslash left|\backslash \; f|\_K\; \backslash in\; L\_p(K),\backslash \; \backslash forall\backslash ,\; K\; \backslash subset\; \backslash Omega,\; K\; \backslash text\{\; compact\}\backslash right.\backslash right\backslash \}.$

An alternative definition, completely analogous to the one given for locally integrable functions, can also be given for locally {{math|p}}-integrable functions: it can also be and proven equivalent to the one in this section.As remarked in the previous section, this is the approach adopted by {{harvtxt|Maz'ya|Shaposhnikova|2009}}, without developing the elementary details. Despite their apparent higher generality, locally {{math|p}}-integrable functions form a subset of locally integrable functions for every {{math|p}} such that {{math|1 < p ≤ +∞}}.Precisely, they form a vector subspace of {{math|L_1,loc(Ω)}}: see {{EquationNote|7|Corollary 1}} to {{EquationNote|6|Theorem 2}}.

Apart from the different glyphs which may be used for the uppercase "L",See for example {{Harv|Vladimirov|2002|p=3}}, where a calligraphic ℒ is used. there are few variants for the notation of the set of locally integrable functions

• $L^p\_\{\backslash mathrm\{loc\}\}(\backslash Omega),$ adopted by {{harv|Hörmander|1990|p=37}}, {{Harv|Strichartz|2003|pp=12–13}} and {{Harv|Vladimirov|2002|p=3}}.
• $L\_\{p,\backslash mathrm\{loc\}\}(\backslash Omega),$ adopted by {{harv|Maz'ya|Poborchi|1997|p=4}} and {{Harvtxt|Maz'ya|Shaposhnikova|2009|p=44}}.
• $L\_p(\backslash Omega,\backslash mathrm\{loc\}),$ adopted by {{harv|Maz'ja|1985|p=6}} and {{harv|Maz'ya|2011|p=2}}.

{{EquationRef|5|Theorem 1}}.See {{harv|Gilbarg|Trudinger|2001|p=147}}, {{harv|Maz'ya|Poborchi|1997|p=5}} for a statement of this results, and also the brief notes in {{harv|Maz'ja|1985|p=6}} and {{harv|Maz'ya|2011|p=2}}. {{math|L_p,loc}} is a complete metrizable space: its topology can be generated by the following metric:

:$d(u,v)=\backslash sum\_\{k\backslash geq\; 1\}\backslash frac\{1\}\{2^k\}\backslash frac\{\backslash Vert\; u\; -\; v\backslash Vert\_\{p,\backslash omega\_k\}\}\{1+\backslash Vert\; u\; -\; v\backslash Vert\_\{p,\backslash omega\_k\}\}\backslash qquad\; u,\; v\backslash in\; L\_\{p,\backslash mathrm\{loc\}\}(\backslash Omega),$

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

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

::$\{\backslash Vert\; u\; \backslash Vert\_\{p,\backslash omega\_k\}\}\; =\; \backslash left\; (\backslash int\_\{\backslash omega\_k\}\; |\; u(x)|^p\; \backslash ,\backslash mathrm\{d\}x\backslash right)^\{1/p\}\backslash qquad\backslash forall\backslash ,\; u\backslash in\; L\_\{p,\backslash mathrm\{loc\}\}(\backslash Omega).$

In references {{harv|Gilbarg|Trudinger|2001|p=147}}, {{harv|Maz'ya|Poborchi|1997|p=5}}, {{harv|Maz'ja|1985|p=6}} and {{harv|Maz'ya|2011|p=2}}, this theorem is stated but not proved on a formal basis:{{harvtxt|Gilbarg|Trudinger|2001|p=147}} and {{harvtxt|Maz'ya|Poborchi|1997|p=5}} only sketch very briefly the method of proof, while in {{harv|Maz'ja|1985|p=6}} and {{harv|Maz'ya|2011|p=2}} it is assumed as a known result, from which the subsequent development starts. a complete proof of a more general result, which includes it, is found in {{harv|Meise|Vogt|1997|p=40}}.

{{EquationRef|6|Theorem 2}}. Every function {{math|f}} belonging to {{math|L_p(Ω)}}, {{math|1 ≤ p ≤ +∞}}, where {{math|Ω}} is an open subset of $\backslash mathbb\{R\}^n$, is locally integrable.

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

:

where

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

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

:

therefore

:$f\backslash in\; L\_\{1,\backslash mathrm\{loc\}\}(\backslash Omega).$

Note that since the following inequality is true

:

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

{{EquationRef|7|Corollary 1}}. Every function $f$ in $L\_\{p,loc\}(\backslash Omega)$,

Note: If $\backslash Omega$ is an open subset of $\backslash mathbb\{R\}^n$ that is also bounded, then one has the standard inclusion $L\_p(\backslash Omega)\; \backslash subset\; L\_1(\backslash Omega)$ which makes sense given the above inclusion $L\_1(\backslash Omega)\backslash subset\; L\_\{1,loc\}(\backslash Omega)$. But the first of these statements is not true if $\backslash Omega$ is not bounded; then it is still true that $L\_p(\backslash Omega)\; \backslash subset\; L\_\{1,loc\}(\backslash Omega)$ for any $p$, but not that $L\_p(\backslash Omega)\backslash subset\; L\_1(\backslash Omega)$. To see this, one typically considers the function $u(x)=1$, which is in $L\_\{\backslash infty\}(\backslash mathbb\{R\}^n)$ but not in $L\_p(\backslash mathbb\{R\}^n)$ for any finite $p$.

{{EquationRef|7|Theorem 3}}. A function {{math|f}} is the density of an absolutely continuous measure if and only if $f\backslash in\; L\_\{1,loc\}$.

The proof of this result is sketched by {{harv|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.According to {{harvtxt|Saks|1937|p=36}}, "If {{math|E}} is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure ({{math|μ}}), then, in order that an additive function of a set ({{math|𝔛}}) on {{math|E}} be absolutely continuous on {{math|E}}, it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of {{math|E}}". Assuming ({{math|μ}}) to be the Lebesgue measure, the two statements can be seen to be equivalent.

• The constant function {{math|1}} defined on the real line is locally integrable but not globally integrable since the real line has infinite measure. More generally, constants, continuous functionsSee for example {{harv|Hörmander|1990|p=37}}. and integrable functions are locally integrable.See {{harv|Strichartz|2003|p=12}}.
• 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} \quad x \in \mathbb R

: is not locally integrable at {{math|x {{=}} 0}}: it is indeed locally integrable near this point since its integral over every compact set not including it is finite. Formally speaking, $1/x\; \backslash in\; L\_\{1,\; loc\}(\backslash mathbb\{R\}\backslash setminus\; 0)$:See {{harv|Schwartz|1998|p=19}}. however, this function can be extended to a distribution on the whole $\backslash mathbb\{R\}$ as a Cauchy principal value.See {{Harv|Vladimirov|2002|pp=19–21}}.

• The preceding example raises a question: does every function which is locally integrable in {{math|Ω}} ⊊ $\backslash mathbb\{R\}$ admit an extension to the whole $\backslash mathbb\{R\}$ 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 $\backslash mathbb\{R\}$.See {{Harv|Vladimirov|2002|p=21}}.

• The following example, similar to the preceding one, is a function belonging to {{math|L_1,loc}}($\backslash mathbb\{R\}$ \ 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 {{math|k₁}} and {{math|k₂}} are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order

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

:Again it does not define any distribution on the whole $\backslash mathbb\{R\}$, if {{math|k₁}} or {{math|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.For a brief discussion of this example, see {{harv|Schwartz|1998|pp=131–132}}.

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.

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

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