Lebesgue integral
2010 Mathematics Subject Classification: Primary: 28A25 [MSN][ZBL]
The most important generalization of the concept of an integral. Let $(X,\mu)$ be a space with a non-negative complete countably-additive measure $\mu$ (cf. Countably-additive set function; Measure space), where $\mu(X)<\infty$. A simple function is a measurable function $g:X\to\mathbb R$ that takes at most a countable set of values: $g(x)=y_n$, $y_n\ne y_k$ for $n\ne k$, if $x\in X_n$, $\bigcup\limits_{n=1}^{\infty}X_n=X$. A simple function $g$ is said to be summable if the series \begin{equation} \sum\limits_{n=1}^{\infty}y_n\mu(X_n) \end{equation} converges absolutely (cf. Absolutely convergent series); the sum of this series is the Lebesgue integral \begin{equation} \int\limits_Xg\,d\mu. \end{equation} A function $f:X\to\mathbb R$ is summable on $X$ (the notation is $f\in L_1(X,\mu)$) if there is a sequence of simple summable functions $g_n$ uniformly convergent (cf. Uniform convergence) to $f$ on a set of full measure, and if the limit \begin{equation} \lim\limits_{n\to\infty}\int\limits_{X}g_n\,d\mu = I \end{equation} is finite. The number $I$ is the Lebesgue integral \begin{equation} \int\limits_Xf\, d\mu. \end{equation}
This is well-defined: the limit exists and does not depend on the choice of the sequence . If , then is a measurable almost-everywhere finite function on . The Lebesgue integral is a linear non-negative functional on with the following properties:
1) if and if
then and
2) if , then and
3) if , and is measurable, then and
4) if and is measurable, then and
In the case when and , , the Lebesgue integral is defined as
under the condition that this limit exists and is finite for any sequence such that , , . In this case the properties 1), 2), 3) are preserved, but condition 4) is violated.
For the transition to the limit under the Lebesgue integral sign see Lebesgue theorem.
If is a measurable set in , then the Lebesgue integral
is defined either as above, by replacing by , or as
where is the characteristic function of ; these definitions are equivalent. If , then for any measurable . If
if is measurable for every , if
and if , then
Conversely, if under these conditions on one has for every and if
then and the previous equality is true (-additivity of the Lebesgue integral).
The function of sets given by
is absolutely continuous with respect to (cf. Absolute continuity); if , then is a non-negative measure that is absolutely continuous with respect to . The converse assertion is the Radon–Nikodým theorem.
For functions the name "Lebesgue integral" is applied to the corresponding functional if the measure is the Lebesgue measure; here, the set of summable functions is denoted simply by , and the integral by
For other measures this functional is called a Lebesgue–Stieltjes integral.
If , and if is a non-decreasing absolutely continuous function, then
If , and if is monotone on , then and there is a point such that
(the second mean-value theorem).
In 1902 H. Lebesgue gave (see [Le]) a definition of the integral for and measure equal to the Lebesgue measure. He constructed simple functions that uniformly approximate almost-everywhere on a set of finite measure a measurable non-negative function , and proved the existence of a common limit (finite or infinite) of the integrals of these simple functions as they tend to . The Lebesgue integral is a basis for various generalizations of the concept of an integral. As N.N. Luzin remarked [Lu], property 2), called absolute integrability, distinguishes the Lebesgue integral for from all possible generalized integrals.
References
[Le] | H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928) MR2857993 Zbl 54.0257.01 |
[Lu] | N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1915) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) |
[KF] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801 |
Comments
For other generalizations of the notion of an integral see -integral; Bochner integral; Boks integral; Burkill integral; Daniell integral; Darboux sum; Denjoy integral; Kolmogorov integral; Perron integral; Perron–Stieltjes integral; Pettis integral; Radon integral; Stieltjes integral; Strong integral; Wiener integral. See also, of course, Riemann integral. See also Double integral; Improper integral; Fubini theorem (on changing the order of integration).
References
[H] | P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802 |
[P] | I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian) MR0264015 Zbl 0206.06401 |
[S] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05 |
[Ro] | H.L. Royden, "Real analysis", Macmillan (1968) |
[Ru] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24 MR1736644 MR1645547 MR0924157 MR0850722 MR0662565 MR0344043 MR0210528 Zbl 1038.00002 Zbl 0954.26001 Zbl 0925.00005 Zbl 0613.26001 Zbl 0925.00003 Zbl 0278.26001 Zbl 0142.01701 |
[HS] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202 |
Lebesgue integral. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lebesgue_integral&oldid=29352