# Locally integrable function

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{{Short description|Function which is integrable on its domain}}
In [mathematics](/source/mathematics), a '''locally integrable function''' (sometimes also called '''locally summable function''')<ref>According to {{harvtxt|Gel'fand|Shilov|1964|p=3}}.</ref> is a [function](/source/function_(mathematics)) which is integrable (so its integral is finite) on every [compact subset](/source/compact_subset) of its [domain of definition](/source/domain_of_definition). The importance of such functions lies in the fact that their [function space](/source/function_space) is similar to [p-integrable function spaces](/source/Lp_space) (<math display=inline>L^p</math> spaces), but its members are not required to satisfy any growth restriction on their behaviour 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===
{{EquationRef|1|Definition 1}}.<ref name="ScVl">See for example {{Harv|Schwartz|1998|p=18}} and {{Harv|Vladimirov|2002|p=3}}.</ref> Let <math display=inline>\Omega</math> be an [open set](/source/open_set) in  the [Euclidean space](/source/Euclidean_space) <math display=inline>\mathbb{R}^n</math> and <math display=inline>f:\Omega\to\Bbb C</math> be a [Lebesgue](/source/Lebesgue_measure) [measurable function](/source/measurable_function). If <math display=inline>f</math> on <math display=inline>\Omega</math> is such that

:<math> \int_K | f |\, \mathrm{d}x <+\infty,</math>

i.e. its [Lebesgue integral](/source/Lebesgue_integral) is finite on all [compact subsets](/source/compact_set) <math display=inline>K</math> of <math display=inline>\Omega</math>,<ref>Another slight variant of this definition, chosen by {{harvtxt|Vladimirov|2002|p=1}}, is to require only that <math display=inline>K\Subset\Omega</math> (or, using the notation of {{harvtxt|Gilbarg|Trudinger|2001|p=9}}, <math display=inline>K\subset\subset\Omega</math>), meaning that <math display=inline>K</math> ''is strictly included in'' <math display=inline>\Omega</math> i.e. it is a set having compact [closure](/source/Closure_(topology)) [strictly included](/source/subset) in the given ambient set.</ref> then <math display=inline>f</math>  is called ''locally integrable''. The [set](/source/Set_(mathematics)) of all such functions is denoted by <math display=inline>L_{1,\text{loc}}(\Omega)</math>:

:<math>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 display=inline>\left.f\right|_K</math> denotes the [restriction](/source/restriction_of_a_function) of <math display=inline>f</math>  to the set <math display=inline>K</math>.

===An alternative definition===
{{EquationRef|2|Definition 2}}.<ref>See for example {{Harv|Strichartz|2003|pp=12–13}}.</ref> Let <math display=inline>\Omega</math> be an open set in the Euclidean space <math display=inline>\mathbb{R}^n</math>. Then a [function](/source/Function_(mathematics)) <math display=inline>f:\Omega\to \mathbb{C}</math> such that

:<math> \int_\Omega | f \varphi|\, \mathrm{d}x <+\infty,</math>

for each [test function](/source/test_function) <math display=inline>\varphi\in C_c^\infty(\Omega)</math> is called ''locally integrable'', and the set of such functions is denoted  by <math display=inline>L_{1,\text{loc}}(\Omega)</math>. Here, <math display=inline>C_c^\infty(\Omega)</math> denotes the set of all infinitely differentiable functions <math display=inline>\varphi\colon\Omega\to\Bbb R</math> with [compact support](/source/Support_(mathematics)) contained in <math display=inline>\Omega</math>.

This definition has its roots in the approach to measure and integration theory based on the concept of a [continuous linear functional](/source/Continuous_linear_functional) on a [topological vector space](/source/topological_vector_space), developed by the [Nicolas Bourbaki](/source/Nicolas_Bourbaki) school.<ref>This approach was praised by {{harvtxt|Schwartz|1998|pp=16–17}} who remarked also its usefulness, however using {{EquationNote|1|Definition&nbsp;1}} to define locally integrable functions.</ref> It is also the one adopted by {{Harvtxt|Strichartz|2003}} and by {{Harvtxt|Maz'ya|Shaposhnikova|2009|p=34}}.<ref>Note that Maz'ya and Shaposhnikova define only the "localized" version of the [Sobolev space](/source/Sobolev_space) <math display=inline>W^{k,p}(\Omega)</math>, nevertheless explicitly asserting that the same method is used to define local versions of all other [Banach space](/source/Banach_space)s used in the cited book. In particular,  <math display=inline>L_{1,\text{loc}}(\Omega)</math> is introduced on page 44.</ref> This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:

{{EquationRef|3|Lemma 1}}. A given function <math display=inline>f:\Omega\to \mathbb{C}</math> is locally integrable according to {{EquationNote|1|Definition&nbsp;1}} if and only if it is locally integrable according to {{EquationNote|2|Definition&nbsp;2}}, i.e.,

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

'''Proof of''' {{EquationNote|4|Lemma&nbsp;1}}

'''Only if part''':  Let <math display=inline>\varphi\in C_c^\infty(\Omega)</math> be a test function. It is [bounded](/source/Extreme_value_theorem) by its [supremum norm](/source/supremum_norm) <math display=inline>\lVert\varphi\rVert_\infty</math>, measurable, and has a [compact support](/source/Support_(mathematics)), let's call it <math display=inline>K</math>. Hence,

:<math>\int_\Omega | f \varphi|\, \mathrm{d}x = \int_K |f|\,|\varphi|\, \mathrm{d}x \le\|\varphi\|_\infty\int_K | f |\, \mathrm{d}x<\infty</math>

by {{EquationNote|1|Definition&nbsp;1}}.

'''If part''': Let <math display=inline>K</math> be a compact subset of the open set <math display=inline>\Omega</math>. We will first construct a test function <math display=inline>\varphi_K\in C_c^\infty(\Omega)</math> which majorises the [indicator function](/source/indicator_function) <math display=inline>\chi_K</math> of <math display=inline>K</math>.
The [usual set distance](/source/Distance)<ref>Not to be confused with the [Hausdorff distance](/source/Hausdorff_distance).</ref> between <math display=inline>K</math> and the [boundary](/source/Boundary_(topology)) <math display=inline>\partial\Omega</math> is strictly greater than zero, i.e.,

:<math>\Delta:=d(K,\partial\Omega)>0,</math>

hence it is possible to choose a [real number](/source/real_number) <math display=inline>\delta</math> such that <math display=inline>\Delta >2\delta>0</math> (if <math display=inline>\partial\Omega</math> is the empty set, take <math display=inline>\Delta =\infty</math>). Let <math display=inline>K_\delta</math> and <math display=inline>K_{2\delta}</math> denote the [closed](/source/Closure_(topology)) [<math display=inline>\delta</math>-neighborhood](/source/Neighbourhood_(mathematics)) and <math display=inline>2\delta</math>-neighborhood of <math display=inline>K</math>, respectively.  They are likewise compact and satisfy

:<math>K\subset K_\delta\subset K_{2\delta}\subset\Omega,\qquad d(K_\delta,\partial\Omega)=\Delta-\delta>\delta>0.</math>

Now use [convolution](/source/convolution) to define the function <math display=inline>\varphi_K : \Omega\to \mathbb{R}</math> by

:<math>\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 <math display=inline>\varphi_\delta</math> is a [mollifier](/source/mollifier) constructed by using the [standard positive symmetric one](/source/Mollifier). Obviously <math display=inline>\varphi_K</math> is non-negative in the sense that <math display=inline>\varphi_K \ge 0</math>, infinitely differentiable, and its support is contained in <math display=inline>K_{2\delta}</math>. In particular, it is a test function. Since <math display=inline>\varphi_K(x)=1</math> for all <math display=inline>x\in K</math>, we have that <math display=inline>\chi_K \le \varphi_K</math>.

Let <math display=inline>f</math> be a locally integrable function according to {{EquationNote|2|Definition&nbsp;2}}. Then

:<math>\int_K|f|\,\mathrm{d}x=\int_\Omega|f|\chi_K\,\mathrm{d}x
\le\int_\Omega|f|\varphi_K\,\mathrm{d}x<\infty.
</math>

Since this holds for every compact subset <math display=inline>K</math> of <math display=inline>\Omega</math>, the function <math display=inline>f</math>  is locally integrable according to {{EquationNote|1|Definition&nbsp;1}}. □

===General definition of local integrability on a generalized measure space===
The classical {{EquationNote|1|Definition&nbsp;1}} of a locally integrable function involves only [measure theoretic](/source/Measure_theory) and [topological](/source/Topological_space)<ref>The notion of compactness must obviously be defined on the given abstract measure space.</ref> concepts and thus can be carried over abstract to [complex-valued](/source/Complex_number) functions on a topological [measure space](/source/measure_space) <math display=inline>(X, \Sigma, \mu)</math>.<ref>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.</ref> Nevertheless, the concept of a locally integrable function can be defined even on a generalised measure space <math display=inline>(X, \mathcal C, \mu)</math>, where <math display=inline>\mathcal C</math> is no longer required to be a [sigma-algebra](/source/%CE%A3-algebra) but only a [ring of sets](/source/ring_of_sets) and, notably, <math display=inline>X</math> does not need to carry the structure of a topological space.

{{EquationRef|8|Definition 1A}}.<ref>{{Harv|Dinculeanu|1966|p=163}}.</ref> Let <math display=inline>(X, \mathcal C, \mu)</math> be an [ordered triple](/source/ordered_triple) where <math display=inline>X</math> is a nonempty set, <math display=inline>\mathcal C</math> is a ring of sets, and <math display=inline>\mu</math> is a [positive measure](/source/positive_measure) on <math display=inline>\mathcal C</math>. Moreover, let <math display=inline>f</math> be a function from <math display=inline>X</math> to a [Banach space](/source/Banach_space) <math display=inline>B</math> or to the [extended real number line](/source/extended_real_number_line) <math display=inline>\overline\mathbb{R}</math>. Then <math display=inline>f</math> is said to be ''locally integrable with respect to'' <math display=inline>\mu</math> if for every set <math display=inline>K\in \mathcal C</math>, the function <math display=inline>f\cdot \chi_K</math> is integrable with respect to <math display=inline>\mu</math>.

The equivalence of {{EquationNote|1|Definition&nbsp;1}} and {{EquationNote|8|Definition&nbsp;1A}} when <math display=inline>X</math> is a topological space can be proven by constructing a ring of sets <math display=inline>\mathcal C</math> from the set <math display=inline>\mathcal K</math> of compact subsets of <math display=inline>X</math> by the following steps. 
# It is evident that <math display=inline>\emptyset\in\mathcal K</math> and, moreover, the operations of union <math display=inline>\cup</math> and intersection <math display=inline>\cap</math> make <math display=inline>\mathcal K</math> a [lattice](/source/Lattice_(order)) with [least upper bound](/source/least_upper_bound) <math display=inline>\vee\equiv\cup</math>  and greatest lower bound <math display=inline>\wedge\equiv\cap</math>.<ref name="Dinculeanu_p.7" >{{Harv|Dinculeanu|1966|p=7}}.</ref>
# The class of sets <math display=inline>\mathcal D</math> defined as <math display=inline>\mathcal D \triangleq \{A\setminus B\mid A,B \in \mathcal K\}</math> is a [semiring of sets](/source/Ring_of_sets)<ref name="Dinculeanu_p.7" /> such that <math display=inline>\mathcal D\supset \mathcal K</math> because of the condition <math display=inline>\emptyset\in\mathcal K</math>.
# The class of sets <math display=inline>\mathcal C</math> defined as <math display=inline>\mathcal C \triangleq \{ \cup_{i=1}^n A_i\mid A_i\in \mathcal D \text{ and } A_i\cap A_j=\emptyset\text{ if }i\neq j\}</math>, i.e., the class formed by finite unions of pairwise disjoint sets of <math display=inline>\mathcal D</math>, is a [ring of sets](/source/ring_of_sets), precisely the minimal one generated by <math display=inline>\mathcal K</math>.<ref name="Dinculeanu_p.8" >{{Harv|Dinculeanu|1966|pp=8−9}}.</ref>

By means of this abstract framework, {{harvtxt|Dinculeanu|1966|pp=163–188}} lists and proves several properties of locally integrable functions. Nevertheless, even if working in this more general framework is possible,  all the definitions and properties presented in the following sections deal explicitly only with this latter important case, since the most common applications of such functions are to [distribution theory](/source/Distribution_(mathematics)) on Euclidean spaces,<ref name="ScVl"/> and thus their domain are invariably subsets of a topological space.

===Generalization: locally ''p''-integrable functions===
{{EquationRef|4|Definition 3}}.<ref name="Vlp3">See for example {{Harv|Vladimirov|2002|p=3}} and {{harv|Maz'ya|Poborchi|1997|p=4}}.</ref> Let <math display=inline>\Omega</math> be an open set in the Euclidean space <math display=inline>\mathbb{R}^n</math> and  <math display=inline>f : \Omega\to\mathbb{C}</math> be a Lebesgue measurable function. If, for a given <math display=inline>p</math> with <math display=inline>1\le p\le+\infty</math>, <math display=inline>f</math> satisfies

:<math> \int_K | f|^p \,\mathrm{d}x <+\infty,</math>

i.e., it belongs to [<math display=inline>L_p(K)</math>](/source/Lp_space) for all [compact subsets](/source/compact_set) <math display=inline>K</math> of <math display=inline>\Omega</math>, then <math display=inline>f</math> is called ''locally'' <math display=inline>p</math>-''integrable'' or also <math display=inline>p</math>-''locally integrable''.<ref name="Vlp3"/> The [set](/source/Set_(mathematics)) of all such functions is denoted by <math display=inline>L_{p,\text{loc}}(\Omega)</math>:

:<math>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 <math display=inline>p</math>-integrable functions: it can also be and proven equivalent to the one in this section.<ref>As remarked in the previous section, this is the approach adopted by {{harvtxt|Maz'ya|Shaposhnikova|2009}}, without developing the elementary details.</ref> Despite their apparent higher generality, locally <math display=inline>p</math>-integrable functions form a subset of locally integrable functions for every <math display=inline>p</math> such that  <math display=inline>1 <p \le +\infty</math>.<ref>Precisely, they form a [vector subspace](/source/vector_subspace) of <math display=inline>L_{p,\text{loc}}(\Omega)</math>: see {{EquationNote|7|Corollary&nbsp;1}} to {{EquationNote|6|Theorem&nbsp;2}}.</ref>

=== Notation ===
Apart from the different [glyph](/source/glyph)s which may be used for the uppercase "L",<ref>See for example {{Harv|Vladimirov|2002|p=3}}, where a calligraphic '''ℒ''' is used.</ref> there are few variants for the notation of the set of locally integrable functions
*<math display=inline>L^p_{\mathrm{loc}}(\Omega),</math> adopted by {{harvtxt|Hörmander|1990|p=37}}, {{Harvtxt|Strichartz|2003|pp=12–13}} and {{Harv|Vladimirov|2002|p=3}}.
*<math display=inline>L_{p,\mathrm{loc}}(\Omega),</math> adopted by {{harvtxt|Maz'ya|Poborchi|1997|p=4}} and {{Harvtxt|Maz'ya|Shaposhnikova|2009|p=44}}.
*<math display=inline>L_p(\Omega,\mathrm{loc}),</math> adopted by {{harvtxt|Maz'ja|1985|p=6}} and {{harvtxt|Maz'ya|2011|p=2}}.

== Properties ==

===''L''<sub>''p'',loc</sub> is a complete metric space for all ''p'' ≥ 1===
{{EquationRef|5|Theorem 1}}.<ref>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}}.</ref>  <math display=inline>L_{p,\text{loc}}</math> is a [complete metrizable space](/source/Complete_metric_space): its topology can be generated by the following [metric](/source/Metric_(mathematics)):
:<math>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  <math display=inline>\{\omega_k\}_{k\ge 1}</math> is a family of non empty open sets such that
* <math display=inline>\omega_k\Subset \omega_{k+ 1}</math>, meaning that <math display=inline>\omega_k</math> ''is compactly contained in'' <math display=inline> \omega_{k+ 1}</math> i.e. each of them is a set whose closure is compact and strictly included in the set of higher index.<ref>In turn this simply means that the boundaries of two sets of the family with different index do not touch.</ref>
* <math display=inline>\cup_{k}\omega_k= \Omega</math> and finally
* <math display=inline> {\Vert\cdot\Vert}_{p,\omega_k}\to\mathbb{R}^+</math>, <math>k\in \mathbb{N}</math> is an [indexed family](/source/indexed_family) of [seminorm](/source/seminorm)s, defined as
::<math> {\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 {{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:<ref>{{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.</ref> a complete proof of a more general result, which includes it, can be found in {{harv|Meise|Vogt|1997|p=40}}.

===''L''<sub>''p''</sub> is a subspace of ''L''<sub>1,loc</sub> for all ''p'' ≥ 1===
{{EquationRef|6|Theorem 2}}. Every function <math display=inline>f</math> belonging to <math display=inline>L_{p,\text{loc}}(\Omega)</math>, <math display=inline>1\le p\le+\infty</math>, where <math display=inline>\Omega</math> is an [open subset](/source/open_subset) of <math display=inline>\mathbb{R}^n</math>, is locally integrable.

'''Proof'''. The case <math display=inline>p=1</math> is trivial, therefore in the sequel of the proof it is assumed that <math display=inline>1< p\le+\infty</math>. Consider the [characteristic function](/source/Indicator_function) <math display=inline>\chi_K</math> of a compact subset <math display=inline>K</math> of <math display=inline>\Omega</math>: then, for <math display=inline>p\le+\infty</math>,

:<math>\left|{\int_\Omega|\chi_K|^q\,\mathrm{d}x}\right|^{1/q}=\left|{\int_K \mathrm{d}x}\right|^{1/q}=|K|^{1/q}<+\infty,</math>

where
*<math display=inline>q</math> is a [positive number](/source/positive_number) such that <math display=inline>1 / p +1 /q =1</math> for a given <math display=inline>1\le p\le+\infty</math>,
*<math display=inline>\vert K\vert</math> is the [Lebesgue measure](/source/Lebesgue_measure) of the [compact set](/source/compact_set) <math display=inline>K</math>.
Then for any <math display=inline>f</math> belonging to <math display=inline>L_{p}(\Omega)</math> the [product](/source/Product_(mathematics)) by <math display=inline>f\chi_K</math> is [integrable](/source/Integrable_function) by [Hölder's inequality](/source/H%C3%B6lder's_inequality) i.e. belongs to <math display=inline>L_{1}(\Omega)</math> and

:<math>{\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}<+\infty,</math>

therefore

:<math>f\in L_{1,\mathrm{loc}}(\Omega).</math>

Note that since the following inequality is true

:<math>{\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}<+\infty,</math>

the theorem is true also for functions <math display=inline>f</math> belonging only to the space of locally <math display=inline>p</math>-integrable functions, therefore the theorem implies also the following result.

{{EquationRef|7|Corollary 1}}. Every function <math display=inline>f</math> in <math display=inline>L_{p,\text{loc}}(\Omega)</math>, <math display=inline>1<p\leq+\infty </math>, is locally integrable, i. e. belongs to <math display=inline>> L_{1,\text{loc}}(\Omega) </math>.

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

=== ''L''<sub>1,loc</sub> is the space of densities of absolutely continuous measures===

{{EquationRef|7|Theorem 3}}. A function <math display=inline>f</math> is the [density](/source/Density_function_(measure_theory)) of an [absolutely continuous measure](/source/Absolute_continuity) if and only if <math>f\in L_{1,\text{loc}}</math>.

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](/source/Radon%E2%80%93Nikodym_theorem) given by [Stanisław Saks](/source/Stanis%C5%82aw_Saks) in his treatise.<ref>According to {{harvtxt|Saks|1937|p=36}}, "''If <math display=inline>E</math> is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure <math display=inline>\mu</math>, then, in order that an additive function of a set <math display=inline>\boldsymbol{\mathfrak X}</math> on <math display=inline>E</math> be absolutely continuous on <math display=inline>E</math>, it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of <math display=inline>E</math>''". Assuming <math display=inline>\mu</math> to be the Lebesgue measure, the two statements can be seen to be equivalent.</ref>

==Examples==
*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](/source/constant_(mathematics)), [continuous function](/source/continuous_function)s<ref>See for example {{harv|Hörmander|1990|p=37}}.</ref> and [integrable function](/source/integrable_function)s are locally integrable.<ref>See {{harv|Strichartz|2003|p=12}}.</ref>
*The function <math display=inline>f(x) = 1/x</math> for <math display=inline>x \in (0, 1)</math> is locally but not globally integrable on <math display=inline>(0, 1)</math>. It is locally integrable since any compact set <math display=inline>K \subset (0, 1)</math> has positive distance from <math display=inline>0</math> and <math display=inline>f</math> is hence bounded on <math display=inline>K</math>. 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>
f(x)=
\begin{cases}
1/x &x\neq 0,\\
0 & x=0,
\end{cases} \quad x \in \mathbb R
</math>
: is not locally integrable at <math display=inline>x= 0</math>: it is indeed locally integrable near this point since its integral over every compact set not including it is finite. Formally speaking, <math display=inline>1/x \in L_{1, loc}(\mathbb{R}\setminus 0)</math>:<ref>See {{harv|Schwartz|1998|p=19}}.</ref> however, this function can be extended to a distribution on the whole <math display=inline>\mathbb{R}</math> as a [Cauchy principal value](/source/Cauchy_principal_value).<ref>See {{Harv|Vladimirov|2002|pp=19–21}}.</ref>
* The preceding example raises a question: does every function which is locally integrable in <math display=inline>\Omega\subsetneq \mathbb{R}</math> admit an extension to the whole <math display=inline>\mathbb{R}</math> as a distribution? The answer is negative, and a counterexample is provided by the following function:
:: <math>
f(x)= 
\begin{cases}
e^{1/x} &x\neq 0,\\
0 & x=0,
\end{cases}
</math>
: does not define any distribution on <math display=inline>\mathbb{R}</math>.<ref>See {{Harv|Vladimirov|2002|p=21}}.</ref>  
* The following example, similar to the preceding one, is a function belonging to <math display=inline>L_{1,\text{loc}}(\mathbb{R}, 0)</math> which serves as an elementary [counterexample](/source/counterexample) in the application of the theory of distributions to [differential operator](/source/differential_operator)s with [irregular singular coefficients](/source/Irregular_singularity):
:: <math>
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}
</math>
:where <math>k_1</math> and <math>k_2</math> are [complex constants](/source/Complex_number), is a general solution of the following elementary [non-Fuchsian differential equation](/source/Fuchsian_differential_equation) of first order 
::<math>x^3\frac{\mathrm{d}f}{\mathrm{d}x}+2f=0.</math>
:Again it does not define any distribution on the whole <math>\mathbb{R}</math>, if <math display=inline>k_1</math> or <math display=inline>k_2</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.<ref>For a brief discussion of this example, see {{harv|Schwartz|1998|pp=131–132}}.</ref>

== Applications ==

Locally integrable functions play a prominent role in [distribution theory](/source/Distribution_(mathematics)) and they occur in the definition of various classes of [functions](/source/function_(mathematics)) and [function space](/source/function_space)s, like [functions of bounded variation](/source/Bounded_variation). Moreover, they appear in the [Radon–Nikodym theorem](/source/Radon%E2%80%93Nikodym_theorem) by characterizing the absolutely continuous part of every measure.

== See also ==
*[Compact set](/source/Compact_set)
*[Distribution (mathematics)](/source/Distribution_(mathematics))
*[Lebesgue's density theorem](/source/Lebesgue's_density_theorem)
*[Lebesgue differentiation theorem](/source/Lebesgue_differentiation_theorem)
*[Lebesgue integral](/source/Lebesgue_integral)
*[Lp space](/source/Lp_space)

==Notes==
{{reflist|29em}}

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== External links ==
*{{MathWorld |title=Locally integrable|author=Rowland, Todd|urlname=LocallyIntegrable}}
*{{springer
 | title=Locally integrable function
 | id= L/l060460
 | last= Vinogradova
 | first=I.A.
 }}

{{PlanetMath attribution|id=4430|title=Locally integrable function}}

{{Lp spaces}}

Category:Measure theory
Category:Integral calculus
Category:Types of functions
Category:Lp spaces

---
Adapted from the Wikipedia article [Locally integrable function](https://en.wikipedia.org/wiki/Locally_integrable_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Locally_integrable_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.