Locally integrable function
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a locally integrable function is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 which is integrable on any compact set of its domain of definition. Their importance lies on the fact that we do not care about their behavior at infinity.

Formal definition

Formally, let be an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 in the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 ℝn and be a Lebesgue
Lebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

 measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

. If the Lebesgue integral of is such that


i.e. it is finite for all compact subsets  in , then is called locally integrable. The set of all such functions is denoted by :

Properties

Theorem. Every function belonging to
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

, , where is an open subset of ℝn is locally integrable.

To see this, consider the characteristic function  of a compact subset  of : then, for


where
  • is the positive number such that for a given
  • is the Lebesgue measure
    Lebesgue measure
    In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

     of the compact set 

Then by Hölder's inequality
Hölder's inequality
In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....

, the product
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...

  is integrable i.e. belongs to and


therefore


Note that since the following inequality is true


the theorem is true also for functions belonging only to
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

 for each compact subset  of .

Examples

  • The constant function defined on the real line is locally integrable but not globally integrable. More generally, continuous function
    Continuous function
    In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

    s and constants are locally integrable.
  • The function
    is not locally integrable near .

    Applications

    Locally integrable functions play a prominent role in distribution theory
    Distribution (mathematics)
    In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

    . Also they occur in the definition of various classes of functions
    Function (mathematics)
    In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

     and function space
    Function space
    In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

    s, like functions of bounded variation
    Bounded variation
    In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...

    .

    See also

    • Compact set
    • Distribution (mathematics)
      Distribution (mathematics)
      In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

    • Lebesgue integral
    • Lebesgue measure
      Lebesgue measure
      In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

    • Lp(Ω) space
      Lp space
      In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

    The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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