View source for Locally integrable function

{{Short description|Function which is integratable on its domain}}
In [[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 (mathematics)|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|{{math|''L''<sup>''p''</sup>}} 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===
{{EquationRef|1|Definition 1}}.<ref name="ScVl">See for example {{Harv|Schwartz|1998|p=18}} and {{Harv|Vladimirov|2002|p=3}}.</ref> Let {{math|Ω}} be an [[open set]] in  the [[Euclidean space]] <math>\mathbb{R}^n</math> and  {{math|''f'' : Ω → <math>\mathbb{C}</math>}} be a [[Lebesgue measure|Lebesgue]] [[measurable function]]. If {{math|''f''}} on {{math|Ω}} is such that

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

i.e. its [[Lebesgue integral]] is finite on all [[compact set|compact subsets]] {{math|''K''}} of {{math|Ω}},<ref>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 (topology)|closure]] [[subset|strictly included]] in the given ambient set.</ref> then {{math|''f''}}  is called ''locally integrable''. The [[Set (mathematics)|set]] of all such functions is denoted by {{math|''L''<sub>1,loc</sub>(Ω)}}:

:<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 of a function|restriction]] of {{math|''f''}}  to the set {{math|''K''}}.

The classical definition of a locally integrable function involves only [[Measure theory|measure theoretic]] and [[Topological space|topological]]<ref>The notion of compactness must obviously be defined on the given abstract measure space.</ref> concepts and can be carried over abstract to [[Complex number|complex-valued]] functions on a topological [[measure space]] {{math|(''X'', Σ, ''μ'')}}:<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> however, since the most common application of such functions is to [[Distribution (mathematics)|distribution theory]] on Euclidean spaces,<ref name="ScVl"/> all the definitions in this and the following sections deal explicitly only with this important case.

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

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

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''<sub>1,loc</sub>(Ω)}}. Here {{math|{{SubSup|C|c|∞}}(Ω)}} denotes the set of all infinitely differentiable functions {{math|''φ'' : Ω → <math>\mathbb{R}</math>}} with [[Support (mathematics)#Compact support|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#Continuous linear functionals|continuous linear functional]] on a [[topological vector space]], developed by the [[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>Be noted that Maz'ya and Shaposhnikova define explicitly only the "localized" version of the [[Sobolev space]] {{math|''W''<sup>''k'',''p''</sup>(Ω)}}, nevertheless explicitly asserting that the same method is used to define localized versions of all other [[Banach space]]s used in the cited book: in particular,  {{math|''L''<sub>''p'',loc</sub>(Ω)}} 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|''f'' : Ω → <math>\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|3|Lemma&nbsp;1}}===
'''If part''':  Let {{math|''φ'' ∈ {{SubSup|C|c|∞}}(Ω)}} be a test function. It is [[Extreme value theorem|bounded]] by its [[supremum norm]] {{math|<nowiki>||</nowiki>''φ''<nowiki>||</nowiki><sub>∞</sub>}}, measurable, and has a [[Support (mathematics)#Compact support|compact support]], let's call it {{math|''K''}}. 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}}.

'''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<sub>K</sub>'' ∈ {{SubSup|C|c|∞}}(Ω)}} which majorises the [[indicator function]] {{math|''χ<sub>K</sub>''}} of {{math|''K''}}.
The [[Distance#Distances between sets and between a point and a set|usual set distance]]<ref>Not to be confused with the [[Hausdorff distance]].</ref> between {{math|''K''}} and the [[Boundary (topology)|boundary]] {{math|∂Ω}} is strictly greater than zero, i.e.

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

hence it is possible to choose a [[real number]] {{math|''δ''}} such that {{math|Δ > 2''δ'' > 0}} (if {{math|∂Ω}} is the empty set, take {{math|Δ {{=}} ∞}}). Let {{math|''K<sub>δ</sub>''}} and {{math|''K''<sub>2''δ''</sub>}} denote the [[Closure (topology)#Closure of a set|closed]] [[Neighbourhood (mathematics)#In a metric space|{{math|''δ''}}-neighborhood]] and {{math|2''δ''}}-neighborhood of {{math|''K''}}, 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]] to define the function {{math|''φ<sub>K</sub>'' : Ω → <math>\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|''φ<sub>δ</sub>''}} is a [[mollifier]] constructed by using the [[Mollifier#Concrete example|standard positive symmetric one]]. Obviously {{math|''φ<sub>K</sub>''}} is non-negative in the sense that {{math|''φ<sub>K</sub>'' ≥ 0}}, infinitely differentiable, and its support is contained in {{math|''K''<sub>2''δ''</sub>}}, in particular it is a test function. Since {{math|''φ<sub>K</sub>''(''x'') {{=}} 1}} for all {{math|''x'' ∈ ''K''}}, we have that {{math|''χ<sub>K</sub>'' ≤ ''φ<sub>K</sub>''}}.

Let {{math|''f''}}  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|''K''}} of {{math|Ω}}, the function {{math|''f''}}  is locally integrable according to {{EquationNote|1|Definition&nbsp;1}}. □

===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|Ω}} be an open set in  the Euclidean space <math>\mathbb{R}^n</math> and  {{math|''f'' : Ω → }}<math>\mathbb{C}</math> be a Lebesgue measurable function. If, for a given {{math|''p''}} with {{math|1 ≤ ''p'' ≤ +∞}}, {{math|''f''}} satisfies

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

i.e., it belongs to {{math|[[Lp space|''L''<sub>''p''</sub>(''K'')]]}} for all [[compact set|compact subsets]] {{math|''K''}} of {{math|Ω}}, then {{math|''f''}} is called ''locally'' {{math|''p''}}-''integrable'' or also {{math|''p''}}-''locally integrable''.<ref name="Vlp3"/> The [[Set (mathematics)|set]] of all such functions is denoted by {{math|''L''<sub>''p'',loc</sub>(Ω)}}:

:<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|''p''}}-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|''p''}}-integrable functions form a subset of locally integrable functions for every {{math|''p''}} such that {{math|1 < ''p'' ≤ +∞}}.<ref>Precisely, they form a [[vector subspace]] of {{math|''L''<sub>1,loc</sub>(Ω)}}: see {{EquationNote|7|Corollary&nbsp;1}} to {{EquationNote|6|Theorem&nbsp;2}}.</ref>

=== Notation ===
Apart from the different [[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>L^p_{\mathrm{loc}}(\Omega),</math> adopted by {{harv|Hörmander|1990|p=37}}, {{Harv|Strichartz|2003|pp=12–13}} and {{Harv|Vladimirov|2002|p=3}}.
*<math>L_{p,\mathrm{loc}}(\Omega),</math> adopted by {{harv|Maz'ya|Poborchi|1997|p=4}} and {{Harvtxt|Maz'ya|Shaposhnikova|2009|p=44}}.
*<math>L_p(\Omega,\mathrm{loc}),</math> adopted by {{harv|Maz'ja|1985|p=6}} and {{harv|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|''L''<sub>''p'',loc</sub>}} is a [[Complete metric space|complete metrizable space]]: its topology can be generated by the following [[Metric (mathematics)|metric]]:
:<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|{''ω''<sub>''k''</sub>}<sub>''k''≥1</sub>}} is a family of non empty open sets such that
* {{math|''ω''<sub>''k''</sub> ⊂⊂ ''ω''<sub>''k''+1</sub>}}, meaning that {{math|''ω''<sub>''k''</sub>}} ''is compactly included in'' {{math|''ω''<sub>''k''+1</sub>}} i.e. it is a set having compact closure strictly included in the set of higher index.
* {{math|∪<sub>''k''</sub>''ω''<sub>''k''</sub> {{=}} Ω}}.
* <math>\scriptstyle{\Vert\cdot\Vert_{p,\omega_k}}\to\mathbb{R}^+</math>, ''k'' ∈ <math>\mathbb{N}</math> is an [[indexed family]] of [[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 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:<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, is 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|''f''}} belonging to {{math|''L''<sub>''p''</sub>(Ω)}}, {{math|1 ≤ ''p'' ≤ +∞}}, where {{math|Ω}} is an [[open subset]] of <math>\mathbb{R}^n</math>, 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 [[Indicator function|characteristic function]] {{math|''χ''<sub>''K''</sub>}} of a compact subset {{math|''K''}} of  {{math|Ω}}: then, for {{math|''p'' ≤ +∞}},

:<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|''q''}} is a [[positive number]] such that {{math|1/''p'' + 1/''q''}} = {{math|1}} for a given {{math|1 ≤ ''p'' ≤ +∞}}
*{{math|<nowiki>|</nowiki>''K''<nowiki>|</nowiki>}} is the [[Lebesgue measure]] of the [[compact set]] {{math|''K''}}
Then for any {{math|''f''}} belonging to {{math|''L''<sub>''p''</sub>(Ω)}}, by [[Hölder's inequality]], the [[Product (mathematics)|product]] {{math|''fχ''<sub>''K''</sub>}} is [[Integrable function|integrable]] i.e. belongs to {{math|''L''<sub>1</sub>(Ω)}} 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|''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 <math> f </math> in <math>L_{p,loc}(\Omega)</math>, <math> 1<p\leq\infty </math>, is locally integrable, i. e. belongs to <math> L_{1,loc}(\Omega) </math>.

'''Note:''' If <math> \Omega </math> is an [[open subset]] of <math> \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,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,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|''f''}} is the [[Density function (measure theory)|density]] of an [[Absolute continuity#Absolute continuity of measures|absolutely continuous measure]] if and only if <math> f\in L_{1,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]] given by [[Stanisław Saks]] in his treatise.<ref>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.</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, [[constant (mathematics)|constants]], [[continuous function]]s<ref>See for example {{harv|Hörmander|1990|p=37}}.</ref> and [[integrable function]]s are locally integrable.<ref>See {{harv|Strichartz|2003|p=12}}.</ref>
*The function <math>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>
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 {{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, <math>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>\mathbb{R}</math> as a [[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|Ω}} ⊊ <math>\mathbb{R}</math> admit an extension to the whole <math>\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>\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|''L''<sub>1,loc</sub>}}(<math>\mathbb{R}</math>&nbsp;\&nbsp;0) which serves as an elementary [[counterexample]] in the application of the theory of distributions to [[differential operator]]s with [[Irregular singularity|irregular singular coefficients]]:
:: <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''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}} are [[Complex number|complex constants]], is a general solution of the following elementary [[Fuchsian differential equation|non-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|''k''<sub>1</sub>}} or {{math|''k''<sub>2</sub>}} 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 (mathematics)|distribution theory]] and they occur in the definition of various classes of [[function (mathematics)|functions]] and [[function space]]s, like [[Bounded variation|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]]

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

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| jfm = 63.0183.05 
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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]]