Locally integrable function

In mathematics, a locally integrable function is a function which is integrable 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 behaviour at the boundary of their domain : 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

. Let be an open set in the Euclidean space and be a Lebesgue measurable function. If on is such that
i.e. its Lebesgue integral is finite on all compact subsets of, then is called locally integrable. The set of all such functions is denoted by :
where denotes the restriction of to the set.

An alternative definition

. Let be an open set in the Euclidean space. Then a function such that
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 a continuous linear functional on a topological vector space, developed by the Nicolas Bourbaki school. It is also the one adopted by and by. 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.,
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,
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 between and the boundary is strictly greater than zero, i.e.,
hence it is possible to choose a real number such that . Let and denote the closed metric space|-neighborhood] and -neighborhood of, respectively. They are likewise compact and satisfy
Now use convolution to define the function by
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
Since this holds for every compact subset of, the function is locally integrable according to. □

General definition of local integrability on a generalized measure space

The classical of a locally integrable function involves only measure theoretic and topological concepts and thus can be carried over abstract to complex-valued functions on a topological measure space. Nevertheless, the concept of a locally integrable function can be defined even on a generalised measure space, where is no longer required to be a sigma-algebra but only a ring of sets and, notably, does not need to carry the structure of a topological space.
. Let be an ordered triple where is a nonempty set, is a ring of sets, and is a positive measure on. Moreover, let be a function from to a Banach space or to the extended real number line. Then is said to be locally integrable with respect to if for every set, the function is integrable with respect to.
The equivalence of and when is a topological space can be proven by constructing a ring of sets from the set of compact subsets of by the following steps.

It is evident that and, moreover, the operations of union and intersection make a lattice with least upper bound and greatest lower bound.
The class of sets defined as is a semiring of sets such that because of the condition.
The class of sets defined as, i.e., the class formed by finite unions of pairwise disjoint sets of, is a ring of sets, precisely the minimal one generated by.

By means of this abstract framework, 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 on Euclidean spaces, and thus their domain are invariably subsets of a topological space.

Generalization: locally ''p''-integrable functions

. Let be an open set in the Euclidean space and be a Lebesgue measurable function. If, for a given with, satisfies
i.e., it belongs to [Lp space|] for all compact subsets of, then is called locally -integrable or also -locally integrable. The set of all such functions is denoted by :
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. Despite their apparent higher generality, locally -integrable functions form a subset of locally integrable functions for every such that .

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

adopted by, and.
adopted by and.
adopted by and.

Properties

''L''_''p'',loc is a complete metric space for all ''p'' ≥ 1

. is a complete metrizable space: its topology can be generated by the following metric:
where is a family of non empty open sets such that

, meaning that is compactly contained in i.e. each of them is a set whose closure is compact and strictly included in the set of higher index.
and finally
, is an indexed family of seminorms, defined as

In,, and, this theorem is stated but not proved on a formal basis: a complete proof of a more general result, which includes it, can be 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, 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,
where

is a positive number such that for a given,
is the Lebesgue measure of the compact set.

Then for any belonging to the product by is integrable by Hölder's inequality i.e. belongs to and
therefore
Note that since the following inequality is true
the theorem is true also for functions belonging only to the space of locally -integrable functions, therefore the theorem implies also the following result.
. Every function in,, is locally integrable, i. e. belongs to.
Note: If is an open subset of that is also bounded, then one has the standard inclusion which makes sense given the above inclusion. But the first of these statements is not true if is not bounded; then it is still true that for any, but not that. To see this, one typically considers the function, which is in but not in for any finite.

''L''_1,loc is the space of densities of absolutely continuous measures

. A function is the density of an absolutely continuous measure if and only if.
The proof of this result is sketched by. 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 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 for is locally but not globally integrable on. It is locally integrable since any compact set has positive distance from and is hence bounded on. 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
The preceding example raises a question: does every function which is locally integrable in admit an extension to the whole as a distribution? The answer is negative, and a counterexample is provided by the following function:
The following example, similar to the preceding one, is a function belonging to which serves as an elementary counterexample in the application of the theory of distributions to differential operators with irregular singular 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.