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 thati.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 thatfor 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, satisfiesi.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 strictly included in i.e. it is a set having compact closure strictly included in the set of higher index.
- .
- , k ∈ ℕ is an indexed family of seminorms, defined as
''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
- is a positive number such that = for a given
- is the Lebesgue measure of the compact set
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 x ∈ is locally but not globally integrable on. It is locally integrable since any compact set K ⊆ 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
- The preceding example raises a question: does every function which is locally integrable in ⊊ ℝ admit an extension to the whole ℝ 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