# 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 $$\sum\limits_{n=1}^{\infty}y_n\mu(X_n)$$ converges absolutely (cf. Absolutely convergent series); the sum of this series is the Lebesgue integral $$\int\limits_Xg\,d\mu.$$ 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 $$\lim\limits_{n\to\infty}\int\limits_{X}g_n\,d\mu = I$$ is finite. The number $I$ is the Lebesgue integral $$\int\limits_Xf\, d\mu.$$

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