# Difference between revisions of "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 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 [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.

How to Cite This Entry:
Lebesgue integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_integral&oldid=29350
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article