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 Lp spaces, but its members are not required to satisfy any growth restriction on their behavior at infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity, but are still manageable in a way similar to ordinary integrable functions.
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Standard definition
Definition 1. Let Ω be an open set in the Euclidean space ℝn and f : Ω → ℂ be a Lebesgue measurable function. If f on Ω is such that
i.e. its Lebesgue integral is finite on all compact subsets K of Ω, then f is called locally integrable. The set of all such functions is denoted by L1,loc(Ω):
where f |K denotes the restriction of f to the set K.
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 (X, Σ, μ): 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
Definition 2. Let Ω be an open set in the Euclidean space ℝn. Then a function f : Ω → ℂ such that
for each test function φ ∈ C ∞
c (Ω) is called locally integrable, and the set of such functions is denoted by L1,loc(Ω). Here C ∞
c (Ω) denotes the set of all infinitely differentiable functions φ : Ω → ℝ with compact support contained in Ω.
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 Strichartz (2003) and by Maz'ya & Shaposhnikova (2009, p. 34). This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves:
Lemma 1. A given function f : Ω → ℂ is locally integrable according to Definition 1 if and only if it is locally integrable according to Definition 2, i.e.
Generalization: locally p-integrable functions
Definition 3. Let Ω be an open set in the Euclidean space ℝn and f : Ω → ℂ be a Lebesgue measurable function. If, for a given p with 1 ≤ p ≤ +∞, f satisfies
i.e., it belongs to Lp(K) for all compact subsets K of Ω, then f is called locally p-integrable or also p-locally integrable. The set of all such functions is denoted by Lp,loc(Ω):
An alternative definition, completely analogous to the one given for locally integrable functions, can also be given for locally p-integrable functions: it can also be and proven equivalent to the one in this section. Despite their apparent higher generality, locally p-integrable functions form a subset of locally integrable functions for every p such that 1 < p ≤ +∞.
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
Lp,loc is a complete metric space for all p ≥ 1
Theorem 1. Lp,loc is a complete metrizable space: its topology can be generated by the following metric:
where {ωk}k≥1 is a family of non empty open sets such that
In references (Gilbarg & Trudinger 1998, 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: a complete proof of a more general result, which includes it, is found in (Meise & Vogt 1997, p. 40).
Lp is a subspace of L1,loc for all p ≥ 1
Theorem 2. Every function f belonging to Lp(Ω), 1 ≤ p ≤ +∞, where Ω is an open subset of ℝn, is locally integrable.
Proof. The case p = 1 is trivial, therefore in the sequel of the proof it is assumed that 1 < p ≤ +∞. Consider the characteristic function χK of a compact subset K of Ω: then, for p ≤ +∞,
where
Then by Hölder's inequality, the product fχK is integrable i.e. belongs to L1(Ω) and
therefore
Note that since the following inequality is true
the theorem is true also for functions f belonging only to the space of locally p-integrable functions, therefore the theorem implies also the following result.
Corollary 1. Every function f in Lp,loc(Ω), 1 < p ≤ +∞, is locally integrable, i. e. belongs to L1,loc(Ω).
L1,loc is the space of densities of absolutely continuous measures
Theorem 3. A function f is the density of an absolutely continuous measure if and only if f ∈L1,loc.
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.
Examples
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.