Lebesgue integral

The integral of a measurable functionMathworldPlanetmath f:X{±} on a measure spaceMathworldPlanetmath (X,𝔅,μ) is usually written

Xf𝑑μ, (1)

although alternative notations such as Xf𝑑x or even f are commonplace.

It is defined via the following steps:

  • If f=𝟏A is the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a set A𝔅, then set

    X𝟏A𝑑μ:=μ(A). (2)
  • If f is a simple functionMathworldPlanetmath (i.e. if f can be written as

    f=k=1nck𝟏Ak,ck (3)

    for some finite collectionMathworldPlanetmath Ak𝔅), then define

    Xf𝑑μ:=k=1nckX𝟏Ak𝑑μ=k=1nckμ(Ak). (4)
  • If f is a nonnegative measurable function (possibly attaining the value at some points), then we define

    Xf𝑑μ:=sup{Xh𝑑μ:h is simple and h(x)f(x) for all xX}. (5)
  • For any measurable function f (possibly attaining the values or - at some points), write f=f+-f- where

    f+:=max(f,0)andf-:=max(-f,0), (6)

    so that |f|=f++f-, and define the integral of f as

    Xf𝑑μ:=Xf+𝑑μ-Xf-𝑑μ, (7)

    provided that Xf+𝑑μ and Xf-𝑑μ are not both .

If μ is Lebesgue measureMathworldPlanetmath and X is any interval in n then the integral is called the Lebesgue integral. If the Lebesgue integral of a function f on a set X exists and is finite (or, equivalently, if X|f|𝑑μ<), then f is said to be Lebesgue integrable. The Lebesgue integral equals the Riemann integral everywhere the latter is defined; the advantage to the Lebesgue integral is that it is often well defined even when the corresponding Riemann integral is undefined. For example, the Riemann integral of the characteristic function of the rationals in [0,1] is undefined, while the Lebesgue integral of this function is simply the measure of the rationals in [0,1], which is 0. Moreover, the conditions under which Lebesgue integrals may be exchanged with each other or with limits or derivativesMathworldPlanetmathPlanetmath, etc., are far less stringent, making the Lebesgue theory a more convenient tool than the Riemann integral for theoretical purposes.

The introduction of the Lebesgue integral was a major advancement in real analysis, soon awakening a large interest in the scientific community. In 1916 Edward Burr Van Vleck, in ”Bulletin of the American Mathematical Society”, vol. 23, wrote: ”This new integral of Lebesgue is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.”

Title Lebesgue integral
Canonical name LebesgueIntegral
Date of creation 2013-03-22 12:18:54
Last modified on 2013-03-22 12:18:54
Owner djao (24)
Last modified by djao (24)
Numerical id 28
Author djao (24)
Entry type Definition
Classification msc 26A42
Classification msc 28A25
Related topic Measure
Related topic LebesgueMeasure
Related topic RiemannIntegral
Related topic RiemannMultipleIntegral
Related topic SimpleFunction
Defines simple function
Defines Lebesgue integrable
Defines integral