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 behavior 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.
The classical definition of a locally integrable function involves only measure theoretic and topological concepts and can be carried over abstract to complex-valued functions on a topological measure space : 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.

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 continuous linear functional on a topological vector space, developed by Nicolas Bourbaki and his 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


Generalization: locally ''p''-integrable functions

. Let be an open set in the Euclidean space ℝn and ℂ 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

''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
In references,, and, this theorem is stated but not proved on a formal basis: 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 ℝn, 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
Then by Hölder's inequality, the product is integrable 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

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.