Locally integrable function

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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 any 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.

Definition[edit]

Standard definition[edit]

Definition 1.[2] Let Ω be an open set in the Euclidean space n and f : Ω → ℂ be a Lebesgue measurable function. If f on Ω is such that

 \int_K | f |\, \mathrm{d}x <+\infty,

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):

L_{1,\mathrm{loc}}(\Omega)=\bigl\{f:\Omega\to\mathbb{C}\text{ measurable}\,\big|\, f|_K \in L_1(K)\ \forall\, K \subset \Omega,\, K \text{ compact}\bigr\},

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[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.

An alternative definition[edit]

Definition 2.[6] Let Ω be an open set in the Euclidean space n. Then a function f : Ω → ℂ such that

 \int_\Omega | f \varphi|\, \mathrm{d}x <+\infty,

for each test function φC 
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:[7] it is also the one adopted by Strichartz (2003) and by Maz'ya & Shaposhnikova (2009, p. 34).[8] 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.

 \int_K | f |\, \mathrm{d}x <+\infty \quad \forall\, K \subset \Omega,\, K \text{ compact} \quad \Longleftrightarrow \quad 
\int_\Omega | f \varphi|\, \mathrm{d}x <+\infty \quad \forall\, \varphi \in C^\infty_{\mathrm{c}}(\Omega).

Generalization: locally p-integrable functions[edit]

Definition 3.[10] 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

 \int_K | f|^p \,\mathrm{d}x <+\infty,

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):

L_{p,\mathrm{loc}}(\Omega)=\left\{f:\Omega\to\mathbb{C}\text{ measurable }\left|\ f\in L_p(K),\ \forall\, K \subset \Omega, K \text{ compact}\right.\right\}.

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]

Notation[edit]

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

Properties[edit]

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

Theorem 1.[14] Lp,loc is a complete metrizable space: its topology can be generated by the following metric:

d(u,v)=\sum_{k\geq 1}\frac{1}{2^k}\frac{\Vert u - v\Vert_{p,\omega_k}}{1+\Vert u - v\Vert_{p,\omega_k}}\qquad u, v\in L_{p,\mathrm{loc}}(\Omega),

where {ωk}k≥1 is a family of non empty open sets such that

  • ωk ⊂⊂ ωk+1, meaning that ωk is strictly included in ωk+1 i.e. it is a set having compact closure strictly included in the set of higher index.
  • kωk = Ω.
  • \scriptstyle{\Vert\cdot\Vert_{p,\omega_k}}\to\mathbb{R}^+, k ∈ ℕ is an indexed family of seminorms, defined as
 {\Vert u \Vert_{p,\omega_k}} = \int_{\omega_k} | u|^p \,\mathrm{d}x\qquad\forall\, u\in L_{p,\mathrm{loc}}(\Omega).

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:[15] 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[edit]

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 ≤ +∞,

\left|{\int_\Omega|\chi_K|^q\,\mathrm{d}x}\right|^{1/q}=\left|{\int_K \mathrm{d}x}\right|^{1/q}=|\mu(K)|^{1/q}<+\infty,

where

Then by Hölder's inequality, the product K is integrable i.e. belongs to L1(Ω) and

{\int_K|f|\,\mathrm{d}x}={\int_\Omega|f\chi_K|\,\mathrm{d}x}\leq\left|{\int_\Omega|f|^p\,\mathrm{d}x}\right|^{1/p}\left|{\int_K \mathrm{d}x}\right|^{1/q}=\|f\|_p|\mu(K)|^{1/q}<+\infty,

therefore

f\in L_{1,\mathrm{loc}}(\Omega).

Note that since the following inequality is true

{\int_K|f|\,\mathrm{d}x}={\int_\Omega|f\chi_K|\,\mathrm{d}x}\leq\left|{\int_K|f|^p \,\mathrm{d}x}\right|^{1/p}\left|{\int_K \mathrm{d}x}\right|^{1/q}=\|f\|_p|\mu(K)|^{1/q}<+\infty,

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[edit]

Theorem 3. A function f is the density of an absolutely continuous measure if and only if fL1,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.[16]

Examples[edit]


f(x)=
\begin{cases}
1/x &x\neq 0\\
0 & x=0
\end{cases}
is not locally integrable in x = 0: it is indeed locally integrable near this point since its integral over any compact set not including it is finite. Formally speaking, 1/xL1,loc(ℝ\0).:[19] however, this function can be extended to a distribution on the whole ℝ as a Cauchy principal value.[20]
  • The preceding example raises a question: does every function which is locally integrable in Ω ⊊ ℝ admit an extension to the whole ℝ as a distribution? The answer is negative, and a counterexample is provided by the following function:

f(x)= 
\begin{cases}
e^{1/x} &x\neq 0\\
0 & x=0
\end{cases}
does not define any distribution on ℝ.[21]

f(x)= 
\begin{cases}
k_1 e^{1/x^2} &x>0\\
0 & x=0\\
k_2 e^{1/x^2} &x<0
\end{cases},
where k1 and k2 are complex constants, is a general solution of the following elementary non-Fuchsian differential equation of first order
x^3\frac{\mathrm{d}f}{\mathrm{d}x}+2f=0.
Again it does not defines any distribution on the whole ℝ, if k1 or k2 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.[22]

Applications[edit]

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.

See also[edit]

Notes[edit]

  1. ^ According to Gel'fand & Shilov (1964, p. 3).
  2. ^ a b See for example (Schwartz 1998, p. 18) and (Vladimirov 2002, p. 3).
  3. ^ Another slight variant of this definition, chosen by Vladimirov (2002, p. 1), is to require only that K ⋐ Ω (or, using the notation of Gilbarg & Trudinger (2001, p. 9), K ⊂⊂ Ω), meaning that K is strictly included in Ω i.e. it is a set having compact closure strictly included in the given ambient set.
  4. ^ The notion of compactness must obviously be defined on the given abstract measure space.
  5. ^ This is the approach developed for example by Cafiero (1959, pp. 285–342) and by Saks (1937, chapter I), without dealing explicitly with the locally integrable case.
  6. ^ See for example (Strichartz 2003, pp. 12–13).
  7. ^ This approach was praised by Schwartz (1998, pp. 16–17) who remarked also its usefulness, however using Definition 1 to define locally integrable functions.
  8. ^ Be noted that Maz'ya and Shaposhnikova define explicitly only the "localized" version of the Sobolev space Wk,p(Ω), nevertheless explicitly asserting that the same method is used to define localized versions of all other Banach spaces used in the cited book: in particular, Lp,loc(Ω) is introduced on page 44.
  9. ^ Not to be confused with the Hausdorff distance.
  10. ^ a b See for example (Vladimirov 2002, p. 3) and (Maz'ya & Poborchi 1997, p. 4).
  11. ^ As remarked in the previous section, this is the approach adopted by Maz'ya & Shaposhnikova (2009), without developing the elementary details.
  12. ^ Precisely, they form a vector subspace of L1,loc(Ω): see Corollary 1 to Theorem 2.
  13. ^ See for example (Vladimirov 2002, p. 3), where a calligraphic is used.
  14. ^ See (Gilbarg & Trudinger 1998, p. 147), (Maz'ya & Poborchi 1997, p. 5) for a statement of this results, and also the brief notes in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2).
  15. ^ Gilbarg & Trudinger (1998, p. 147) and Maz'ya & Poborchi (1997, p. 5) only sketch very briefly the method of proof, while in (Maz'ja 1985, p. 6) and (Maz'ya 2011, p. 2) it is assumed as a known result, from which the subsequent development starts.
  16. ^ According to Saks (1937, p. 36), "If E is a set of finite measure, or, more generally the sum of a sequence of sets of finite measure (μ), then, in order that an additive function of a set (𝔛) on E be absolutely continuous on E, it is necessary and sufficient that this function of a set be the indefinite integral of some integrable function of a point of E". Assuming (μ) to be the Lebesgue measure, the two statements can be seen to be equivalent.
  17. ^ See for example (Hörmander 1990, p. 37).
  18. ^ See (Strichartz 2003, p. 12).
  19. ^ See (Schwartz 1998, p. 19).
  20. ^ See (Vladimirov 2002, pp. 19–21).
  21. ^ See (Vladimirov 2002, p. 21).
  22. ^ For a brief discussion of this example, see (Schwartz 1998, pp. 131–132).

References[edit]

External links[edit]

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