View source for Lebesgue integral

{{MSC|28A25}}

[[Category:Classical measure theory]]

[[Category:TeX wanted]]

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_X g\ 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_X f\ d\mu.
\end{equation}

This is well-defined: the limit $l$ exists and does not depend on the choice of the sequence $g_n$. If $f\in L_1(X,\mu)$, then $f$ is a measurable almost-everywhere finite function on $X$. The Lebesgue integral is a linear non-negative functional on $L_1(X,\mu)$ with the following properties:

1) if $L_1(X,\mu)$ and if

\begin{equation}\mu\{x\in X:\ f(x)\neq h(x)\}=0,\end{equation}

then $h\in L_1(X,\mu)$ and

\begin{equation}\int\limits_X f\ d\mu=\int\limits_X g\ d\mu\end{equation}

2) if $f\in L_1(X,\mu)$, then $|f|\in L_1(X,\mu)$ and

\begin{equation}\left|\int\limits_X f\ d\mu\right|\leq\int\limits_X |f|\ d\mu\end{equation}

3) if $f\in L_1(X,\mu),|h|\leq f$ and $h$ is measurable, then $h\in L_1(X,\mu)$ and

\begin{equation}\left|\int\limits_X h\ d\mu\right|\leq\int\limits_X f\ d\mu\end{equation}

4) if $m\leq f\leq M$ and $f$ is measurable, then $f\in L_1(X,\mu)$ and

\begin{equation}m\mu X\leq\int\limits_X f\ d\mu\leq M\mu X\end{equation}

In the case when $\mu X=+\infty$ and $X=\cup_{n=1}^\infty X_n,\mu X_n<+\infty$ the Lebesgue integral is defined as

\begin{equation}\lim\limits_{n\to\infty}\int\limits_{E_n} f\ du\end{equation}

under the condition that this limit exists and is finite for any sequence $E_n$ such that $\mu E_n<+\infty,E_n\subset E_{n+1},\cup_{n=1}^\infty E_n=X$. 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 $A$ is a [[Measurable set|measurable set]] in $X$, then the Lebesgue integral

\begin{equation}\int\limits_A f\ d\mu\end{equation}

is defined either as above, by replacing $X$ by $A$, or as

\begin{equation}\int\limits_X f\chi_A\ d\mu\end{equation}

where $\chi_A$ is the characteristic function of $A$; these definitions are equivalent. If $f\in L_1(A,\mu)$, then $f\in L_1(A_1,\mu)$ for any measurable $A_1\subset A$.

If

\begin{equation}A=\bigcup_{n=1}^\infty A_n\end{equation}

if $A$ is measurable for every $n$, if

\begin{equation}A_n\cap A_k\ \text{for}\ n\neq k\end{equation}

and if $f\in L_1(A,\mu)$ then

\begin{equation}\int\limits_A f\ d\mu=\sum_{n=1}^\infty \int\limits_{A_n} f\ d\mu\end{equation}

Conversely, if under these conditions on $A_n$ one has $f\in L_1(A,\mu)$ for every $n$ and if $\sum_{n=1}^\infty\int\limits_{A_n} |f|\ d\mu$ then $f\in L_1(A,\mu)$ and the previous equality is true ($\sigma$-additivity of the Lebesgue integral).

The function of sets $A\subset X$ given by $F(A)=\int\limits_A f\ d\mu$ is absolutely continuous with respect to $\mu$ (cf. [[Absolute continuity|Absolute continuity]]); if $f\geq 0$, then $F$ is a non-negative measure that is absolutely continuous with respect to $\mu$. 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}}
|}