Definition
Standard definition
Definition 1. [2] 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
, [3] 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
Definition 2. [4] 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 the Nicolas Bourbaki school: [5] it is also the one adopted by Strichartz (2003) and by Maz'ya & Shaposhnikova (2009 , p. 34). [6] This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:
Lemma 1. A given function
is locally integrable according to Definition 1 if and only if it is locally integrable according to Definition 2 , i.e.

Proof of Lemma 1
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 Definition 1 .
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 [7] between
and the boundary
is strictly greater than zero, i.e.

hence it is possible to choose a real number
such that
(if
is the empty set, take
). Let
and
denote the closed
-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 Definition 2 . Then

Since this holds for every compact subset
of
, the function
is locally integrable according to Definition 1 . □
General definition of local integrability on a generalized measure space
The classical Definition 1 of a locally integrable function involves only measure theoretic and topological [8] concepts and thus can be carried over abstract to complex-valued functions on a topological measure space
: [9] 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 tribe but only a clan and, notably,
does not need to carry a structure of topological space.
Definition 1A. [10] Let
be a ordered triple where
is a nonempty set,
is a clan and
is a positive measure on
. Moreover let
be a function from
to a Banach space
or to the extended real number line
:
is said to be locally integrable respect to
if for every set
the function
is integrable tout court respect to
.
The equivalence of Definition 1 and Definition 1A , when
is a topological space, can be proved by constructing a clan
from the set
of compact subsets of
by the following steps.
- It is evident that
and moreover the inner operations of union
and intersection
make
trivially a lattice with least upper bound
and greatest lower bound
. [11] . - The class of sets
defined as
is a semi-clan, [11] such that
because of condition
. - The class of sets
defined as
i.e. the class formed by finite unions of pairwise disjoint sets of
, is a clan, precisely the minimal one generated by
. [12]
By means of this abstract framework, Dinculeanu (1966 , pp. 163–188) lists and proves several properties of locally integrable functions: nevertheless, even if working in this more general framework is possible, since the most commonly seen applications of such functions is to distribution theory on Euclidean spaces, [2] all the definitions and properties presented in the following sections deal explicitly only with this latter important case.
Generalization: locally p-integrable functions
Definition 3. [13] 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. [13] 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. [14] Despite their apparent higher generality, locally
-integrable functions form a subset of locally integrable functions for every
such that
. [15]
Notation
Apart from the different glyphs which may be used for the uppercase "L", [16] there are few variants for the notation of the set of locally integrable functions
adopted by ( Hörmander 1990 , p. 37), ( Strichartz 2003 , pp. 12–13) and ( Vladimirov 2002 , p. 3).
adopted by ( Maz'ya & Poborchi 1997 , p. 4) and Maz'ya & Shaposhnikova (2009 , p. 44).
adopted by ( Maz'ja 1985 , p. 6) and ( Maz'ya 2011 , p. 2).
Properties
Lp,loc is a complete metric space for all p ≥ 1
Theorem 1. [17]
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. [18]
and finally
,
is an indexed family of seminorms, defined as

In ( Gilbarg & Trudinger 2001 , p. 147), ( Maz'ya & Poborchi 1997 , p. 5), ( Maz'ja 1985 , p. 6) and ( Maz'ya 2011 , p. 2), this theorem is stated but not proved on a formal basis: [19] a complete proof of a more general result, which includes it, can be found in ( Meise & Vogt 1997 , p. 40).
Lp is a subspace of L1,loc for all p ≥ 1
Theorem 2. 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.
Corollary 1. 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
.
L1,loc is the space of densities of absolutely continuous measures
Theorem 3. A function
is the density of an absolutely continuous measure if and only if
.
The proof of this result is sketched by ( Schwartz 1998 , p. 18). 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. [20]