Lebesgue integral

2020 Mathematics Subject Classification: Primary: 28A25 [MSN][ZBL]

The most important generalization of the concept of an integral. Let be a space with a non-negative complete countably-additive measure (cf. Countably-additive set function; Measure space), where . A simple function is a measurable function that takes at most a countable set of values: , for , if , . A simple function is said to be summable if the series

converges absolutely (cf. Absolutely convergent series); the sum of this series is the Lebesgue integral

A function is summable on , , if there is a sequence of simple summable functions uniformly convergent (cf. Uniform convergence) to on a set of full measure, and if the limit

is finite. The number is the Lebesgue integral

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 [1]) 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 [2], property 2), called absolute integrability, distinguishes the Lebesgue integral for from all possible generalized integrals.

References

[1]	H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928)
[2]	N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1915) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)
[3]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)

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

[a1]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[a2]	I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian)
[a3]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[a4]	H.L. Royden, "Real analysis" , Macmillan (1968)
[a5]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24
[a6]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)