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Wikipedia / Locally integrable function

In mathematics, a locally integrable function (sometimes also called locally summable function) 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 p-integrable function spaces spaces), but its members are not required to satisfy any growth restriction on their behaviour 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

. be an open set in the Euclidean space be a Lebesgue measurable function is such that

i.e. its Lebesgue integral is finite on all compact subsets, is called locally integrable. The set:

denotes the restriction.

An alternative definition

.. Then a function such that

for each test function is called locally integrable with compact support.

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:

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

Proof of

Only if part be a test function. It is bounded by its supremum norm, measurable, and has a compact support. Hence,

by .

If part which majorises the indicator function. The usual set distance and the boundary is strictly greater than zero, i.e.,

hence it is possible to choose a real number denote the closed \delta-neighborhood, respectively. They are likewise compact and satisfy

Now use convolution by

is a mollifier constructed by using the standard positive symmetric one.

be a locally integrable function according to . Then

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 to abstract complex-valued functions on a topological measure space. is no longer required to be a sigma-algebra but only a ring of sets does not need to carry the structure of a topological space.

. be an ordered triple is a positive measure to a Banach space or to the extended real number line is said to be locally integrable with respect to.

The equivalence of and by the following steps.

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 without any need of defining a topological structure, 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

. satisfies

i.e., it belongs to L_p(K) for all compact subsets is called locally-integrable-locally integrable. The set:

-integrable functions: it can also be and proven equivalent to the one in this section..

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

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

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

is a family of non empty open sets such 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 .

Lp is a subspace of L1,loc for all p ≥ 1

is an open subset, is locally integrable.

Proof. Consider the characteristic function,

where

is a positive number,
is the Lebesgue measure of the compact set.

the product is integrable by Hölder's inequality and

therefore

Note that since the following inequality is true

-integrable functions, therefore the theorem implies also the following result.

Note: is an open subset.

L1,loc is the space of densities of absolutely continuous measures

is the density of an absolutely continuous measure.

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.
. 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
: as a Cauchy principal value.
as a distribution? The answer is negative, and a counterexample is provided by the following function:
.
which serves as an elementary counterexample in the application of the theory of distributions to differential operators with irregular singular coefficients:
are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order
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.

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.

Notes

References

. Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
. Translated from the original 1958 Russian edition by Eugene Saletan, this is an important monograph on the theory of generalized functions, dealing both with distributions and analytic functionals.
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(available also as ).
(available also as ).
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. English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach: the Mathematical Reviews number refers to the Dover Publications 1964 edition, which is basically a reprint.
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. A monograph on the theory of generalized functions written with an eye towards their applications to several complex variables and mathematical physics, as is customary for the Author.