In mathematics, a locally integrable function (sometimes also called locally summable function)[1] 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 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.
[2] Let Ω be an open set in the Euclidean space
If f on Ω is such that i.e. its Lebesgue integral is finite on all compact subsets K of Ω,[3] then f is called locally integrable.
The set of all such functions is denoted by L1,loc(Ω): where
denotes the restriction of f to the set K. The classical definition of a locally integrable function involves only measure theoretic and topological[4] concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ):[5] however, since the most common application of such functions is to distribution theory on Euclidean spaces,[2] all the definitions in this and the following sections deal explicitly only with this important case.
[6] Let Ω be an open set in the Euclidean space
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).
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:[7] it is also the one adopted by Strichartz (2003) and by Maz'ya & Shaposhnikova (2009, p. 34).
If part: Let φ ∈ C ∞c (Ω) be a test function.
It is bounded by its supremum norm ||φ||∞, measurable, and has a compact support, let's call it K. Hence by Definition 1.
Only if part: Let K be a compact subset of the open set Ω.
We will first construct a test function φK ∈ C ∞c (Ω) which majorises the indicator function χK of K. The usual set distance[9] between K and the boundary ∂Ω is strictly greater than zero, i.e. hence it is possible to choose a real number δ such that Δ > 2δ > 0 (if ∂Ω is the empty set, take Δ = ∞).
They are likewise compact and satisfy Now use convolution to define the function φK : Ω →
by where φδ is a mollifier constructed by using the standard positive symmetric one.
Obviously φK is non-negative in the sense that φK ≥ 0, infinitely differentiable, and its support is contained in K2δ, in particular it is a test function.
Let f be a locally integrable function according to Definition 2.
Then Since this holds for every compact subset K of Ω, the function f is locally integrable according to Definition 1.
[10] Let Ω be an open set in the Euclidean space
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.
[10] 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.
[11] Despite their apparent higher generality, locally p-integrable functions form a subset of locally integrable functions for every p such that 1 < p ≤ +∞.
[12] Apart from the different glyphs which may be used for the uppercase "L",[13] there are few variants for the notation of the set of locally integrable functions Theorem 1.
[14] 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 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:[15] a complete proof of a more general result, which includes it, is found in (Meise & Vogt 1997, p. 40).
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 for any f belonging to Lp(Ω), 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.
A function f is the density of an absolutely continuous measure if and only if
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.
Moreover, they appear in the Radon–Nikodym theorem by characterizing the absolutely continuous part of every measure.
This article incorporates material from Locally integrable function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.