Locally integrable function explained

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 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

R ^{n} and be a Lebesgue measurable function. If on is such that\int _{K} | f | d x < +inf ty,

i.e. its Lebesgue integral is finite on all compact subsets of,^[3] then is called locally integrable. The set of all such functions is denoted by :

L _{1,loc} (\Omega) = l \{f\colon \Omega\to C measurable : f| _{K} \in L _{1(K) \forall} K \subset \Omega, K compact r \},

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

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 :^[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

.^[6] Let be an open set in the Euclidean space

R ^{n} . Then a function such that\int _{\Omega} | f \varphi| d x < +inf ty,

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

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 and by .^[8] This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:

. A given function is locally integrable according to if and only if it is locally integrable according to, i.e.

\int _{K} | f | d x < +inf ty \forall K \subset \Omega, K compact \Longleftrightarrow \int _{\Omega} | f \varphi| d x < +inf ty \forall \varphi \in C ^{inf ty} _{c} (\Omega).

Proof of

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

\int _{\Omega} | f \varphi| d x = \int _{K} |f||\varphi| d x \le\|\varphi\| _{infty\int} _{K} | f | d x< inf ty

by .

Only if part: Let be a compact subset of the open set . We will first construct a test function which majorises the indicator function of .The usual set distance^[9] between and the boundary is strictly greater than zero, i.e.

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

hence it is possible to choose a real number such that (if is the empty set, take). Let and denote the closed -neighborhood and -neighborhood of, respectively. They are likewise compact and satisfy

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

Now use convolution to define the function by

\varphi _{K(x)={\chi} ^{K _{\delta}} \ast\varphi _{\delta(x)}= \int} ^{R ^{n}} \chi _{K _{\delta}} (y)\varphi _{\delta(x-y)dy,}

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

Let be a locally integrable function according to . Then

\int _{K|f|dx=\int} _{\Omega|f|\chi} _{Kdx \le\int} _{\Omega|f|\varphi} _{Kdx<infty.}

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

Generalization: locally p-integrable functions

.^[10] Let be an open set in the Euclidean space

R ^{n} and C be a Lebesgue measurable function. If, for a given with, satisfies\int _{K} | f| ^{p} d x < +inf ty,

i.e., it belongs to for all compact subsets of, then is called locally -integrable or also -locally integrable.^[10] The set of all such functions is denoted by :

L _{p,loc} (\Omega) = \left\{f:\Omega\to C measurable \left| f| _{K} \in L _{p(K), \forall} K \subset \Omega, K compact \right.\right\}.

An alternative definition, completely analogous to the one given for locally integrable functions, can also be given for locally -integrable functions: it can also be and proven equivalent to the one in this section.^[11] Despite their apparent higher generality, locally -integrable functions form a subset of locally integrable functions for every such that .^[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

L ^{p} _{loc} (\Omega), adopted by, and .L _{p,loc} (\Omega), adopted by and .L _{p(\Omega,loc),} adopted by and .

Properties

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

.^[14] is a complete metrizable space: its topology can be generated by the following metric:

d(u,v) = \sum _{k\geq} (1)/(2 ^{k}) (\Vert u - v\Vert _{p,\omega _{k}})/(1+ \Vert u - v\Vert _{p,\omega _{k}}) u, v\in L _{p,loc} (\Omega), where is a family of non empty open sets such that

• , meaning that is compactly included in i.e. it is a set having compact closure strictly included in the set of higher index.
• .

\scriptstyle{\Vert ⋅ \Vert _{p,\omega _{k}} }\to\mathbb^+, k ∈ N is an indexed family of seminorms, defined as{\Vert u \Vert _{p,\omega _{k}} } = \left (\int_ | u(x)|^p \,\mathrmx\right)^\qquad\forall\, u\in L_(\Omega).

In references,, and, 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 .

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

. Every function belonging to,, where is an open subset of

R ^{n} , is locally integrable.

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

\left|{\int _{\Omega|\chi} ^{q d x}\right|} ^{K|} ^{1/q} = \left|{\int _{K} d x}\right| ^{1/q} = |K| ^{1/q} < +inf ty,

where

• is a positive number such that = for a given

Notes and References

1. According to .
2. See for example and .
3. Another slight variant of this definition, chosen by, is to require only that (or, using the notation of,), meaning that is strictly included in 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 and by, without dealing explicitly with the locally integrable case.
6. See for example .
7. This approach was praised by who remarked also its usefulness, however using to define locally integrable functions.
8. Be noted that Maz'ya and Shaposhnikova define explicitly only the "localized" version of the Sobolev space, 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, is introduced on page 44.
9. Not to be confused with the Hausdorff distance.
10. See for example and .
11. As remarked in the previous section, this is the approach adopted by, without developing the elementary details.
12. Precisely, they form a vector subspace of : see to .
13. See for example, where a calligraphic ℒ is used.
14. See, for a statement of this results, and also the brief notes in and .
15. and only sketch very briefly the method of proof, while in and it is assumed as a known result, from which the subsequent development starts.