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[[Category:Classical measure theory]]

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The most important generalization of the concept of an [[Integral|integral]]. Let $(X,\mu)$ be a space with a non-negative complete countably-additive measure $\mu$ (cf. [[Countably-additive set function|Countably-additive set function]]; [[Measure space|Measure space]]), where $\mu(X)<\infty$. A simple function is a [[Measurable function|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{{Anchor|series}}
\begin{equation}
\sum\limits_{n=1}^{\infty}y_n\mu(X_n)
\end{equation}
converges absolutely (cf. [[Absolutely convergent series|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786021.png" /> exists and does not depend on the choice of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786024.png" /> is a measurable almost-everywhere finite function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786025.png" />. The Lebesgue integral is a linear non-negative functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786026.png" /> with the following properties:

1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786027.png" /> and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786028.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786029.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786030.png" /></td> </tr></table>

2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786032.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786033.png" /></td> </tr></table>

3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786036.png" /> is measurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786037.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786038.png" /></td> </tr></table>

4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786040.png" /> is measurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786041.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786042.png" /></td> </tr></table>

In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786045.png" />, the Lebesgue integral is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786046.png" /></td> </tr></table>

under the condition that this limit exists and is finite for any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786050.png" />. 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|Lebesgue theorem]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786051.png" /> is a [[Measurable set|measurable set]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786052.png" />, then the Lebesgue integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786053.png" /></td> </tr></table>

is defined either as above, by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786054.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786055.png" />, or as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786056.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786057.png" /> is the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786058.png" />; these definitions are equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786060.png" /> for any measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786061.png" />. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786062.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786063.png" /> is measurable for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786064.png" />, if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786065.png" /></td> </tr></table>

and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786066.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786067.png" /></td> </tr></table>

Conversely, if under these conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786068.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786069.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786070.png" /> and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786071.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786072.png" /> and the previous equality is true (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786073.png" />-additivity of the Lebesgue integral).

The function of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786074.png" /> given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786075.png" /></td> </tr></table>

is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786076.png" /> (cf. [[Absolute continuity|Absolute continuity]]); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786077.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786078.png" /> is a non-negative measure that is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786079.png" />. The converse assertion is the [[Radon–Nikodým theorem|Radon–Nikodým theorem]].

For functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786080.png" /> the name "Lebesgue integral" is applied to the corresponding functional if the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786081.png" /> is the [[Lebesgue measure|Lebesgue measure]]; here, the set of summable functions is denoted simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786082.png" />, and the integral by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786083.png" /></td> </tr></table>

For other measures this functional is called a [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes integral]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786085.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786086.png" /> is a non-decreasing absolutely continuous function, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786087.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786089.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786090.png" /> is monotone on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786091.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786092.png" /> and there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786093.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786094.png" /></td> </tr></table>

(the second mean-value theorem).

In 1902 H. Lebesgue gave (see {{Cite|Le}}) a definition of the integral for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786095.png" /> and measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786096.png" /> equal to the Lebesgue measure. He constructed simple functions that uniformly approximate almost-everywhere on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786097.png" /> of finite measure a measurable non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786098.png" />, and proved the existence of a common limit (finite or infinite) of the integrals of these simple functions as they tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786099.png" />. The Lebesgue integral is a basis for various generalizations of the concept of an integral. As N.N. Luzin remarked {{Cite|Lu}}, property 2), called absolute integrability, distinguishes the Lebesgue integral for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l057860100.png" /> from all possible generalized integrals.

====References====
{|
|valign="top"|{{Ref|Le}}|| H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928) {{MR|2857993}} {{ZBL|54.0257.01}}
|-
|valign="top"|{{Ref|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)
|-
|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian) {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{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 [[A-integral|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l057860101.png" />-integral]]; [[Bochner integral|Bochner integral]]; [[Boks integral|Boks integral]]; [[Burkill integral|Burkill integral]]; [[Daniell integral|Daniell integral]]; [[Darboux sum|Darboux sum]]; [[Denjoy integral|Denjoy integral]]; [[Kolmogorov integral|Kolmogorov integral]]; [[Perron integral|Perron integral]]; [[Perron–Stieltjes integral|Perron–Stieltjes integral]]; [[Pettis integral|Pettis integral]]; [[Radon integral|Radon integral]]; [[Stieltjes integral|Stieltjes integral]]; [[Strong integral|Strong integral]]; [[Wiener integral|Wiener integral]]. See also, of course, [[Riemann integral|Riemann integral]]. See also [[Double integral|Double integral]]; [[Improper integral|Improper integral]]; [[Fubini theorem|Fubini theorem]] (on changing the order of integration).

====References====
{|
|valign="top"|{{Ref|H}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
|-
|valign="top"|{{Ref|P}}|| I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian) {{MR|0264015}} {{ZBL|0206.06401}}
|-
|valign="top"|{{Ref|S}}|| S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}} {{ZBL|63.0183.05}}
|-
|valign="top"|{{Ref|Ro}}|| H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan (1968)
|-
|valign="top"|{{Ref|Ru}}|| W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24 {{MR|1736644}} {{MR|1645547}} {{MR|0924157}} {{MR|0850722}} {{MR|0662565}} {{MR|0344043}} {{MR|0210528}} {{ZBL|1038.00002}} {{ZBL|0954.26001}} {{ZBL|0925.00005}} {{ZBL|0613.26001}} {{ZBL|0925.00003}} {{ZBL|0278.26001}} {{ZBL|0142.01701}}
|-
|valign="top"|{{Ref|HS}}|| E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}}
|}