# Lebesgue integral

 $\int_{X}f\ d\mu,$ (1)

although alternative notations such as $\int_{X}f\ dx$ or even $\int f$ are commonplace.

It is defined via the following steps:

If $\mu$ is Lebesgue measure  and $X$ is any interval in $\mathbb{R}^{n}$ 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 $\int_{X}|f|\ d\mu<\infty$), 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 derivatives   , 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