In
mathematics, a locally integrable function (sometimes also called locally summable function) is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
which is integrable (so its integral is finite) on every
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
of its
domain of definition. The importance of such functions lies in the fact that their
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vec ...
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
.
[See for example and .] Let be an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
in the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
and be a
Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
. If on is such that
:
i.e. its
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
concepts and can be carried over abstract to
complex-valued functions on a topological
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
: 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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
such that
:
for each
test function
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
contained in .
This definition has its roots in the approach to measure and integration theory based on the concept of
continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
on a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
, developed by the
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
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
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
, measurable, and has a
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
, 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
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
of .
The
usual set distance between and the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
* Boundary (cricket), the edge of the pl ...
is strictly greater than zero, i.e.
:
hence it is possible to choose a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
such that (if is the empty set, take ). Let and denote the
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
-neighborhood and -neighborhood of , respectively. They are likewise compact and satisfy
:
Now use
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
to define the function by
:
where is a
mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) fu ...
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 . □
Generalization: locally ''p''-integrable functions
.
[See for example and .] 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 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
glyph
A glyph () is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A g ...
s 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
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
:
:
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.
* .
*
, ''k'' ∈
is an
indexed family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, whe ...
of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s, defined as
::
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
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
of
, is locally integrable.
Proof. The case is trivial, therefore in the sequel of the proof it is assumed that . Consider the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
of a compact subset of : then, for ,
:
where
* is a
positive number
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
such that = for a given
* is the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides w ...
of the
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
Then for any belonging to , by
Hölder's inequality, the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
is
integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
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
,