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[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
 
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The most important generalization of the concept of an
[[Category:TeX wanted]]
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[[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]];
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}}
+
[[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}
 
\begin{equation}
 
\sum\limits_{n=1}^{\infty}y_n\mu(X_n)
 
\sum\limits_{n=1}^{\infty}y_n\mu(X_n)
 
\end{equation}
 
\end{equation}
converges absolutely (cf. [[Absolutely convergent series|Absolutely convergent series]]); the sum of this series is the Lebesgue integral
+
converges absolutely (cf.
 +
[[Absolutely convergent series|Absolutely convergent series]]); the sum of this series is the Lebesgue integral
 
\begin{equation}
 
\begin{equation}
\int\limits_Xg\,d\mu.
+
\int\limits_X g\ d\mu.
 
\end{equation}
 
\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
+
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}
 
\begin{equation}
\lim\limits_{n\to\infty}\int\limits_{X}g_n\,d\mu = I
+
\lim\limits_{n\to\infty}\int\limits_X g_n\ d\mu = I
 
\end{equation}
 
\end{equation}
 
is finite. The number $I$ is the Lebesgue integral
 
is finite. The number $I$ is the Lebesgue integral
 
\begin{equation}
 
\begin{equation}
\int\limits_Xf\, d\mu.
+
\int\limits_X f\ d\mu.
 
\end{equation}
 
\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:
+
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 <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
+
1) if $L_1(X,\mu)$ 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/l05786038.png" /></td> </tr></table>
+
\begin{equation}\mu\{x\in X:\ f(x)\neq h(x)\}=0,\end{equation}
  
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
+
then $h\in L_1(X,\mu)$ 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>
+
\begin{equation}\int\limits_X f\ d\mu=\int\limits_X g\ d\mu\end{equation}
  
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
+
2) if $f\in L_1(X,\mu)$, then $|f|\in L_1(X,\mu)$ 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/l05786046.png" /></td> </tr></table>
+
\begin{equation}\left|\int\limits_X f\ d\mu\right|\leq\int\limits_X |f|\ d\mu\end{equation}
  
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.
+
3) if $f\in L_1(X,\mu),|h|\leq f$ and $h$ is measurable, then $h\in L_1(X,\mu)$ and
  
For the transition to the limit under the Lebesgue integral sign see [[Lebesgue theorem|Lebesgue theorem]].
+
\begin{equation}\left|\int\limits_X h\ d\mu\right|\leq\int\limits_X f\ d\mu\end{equation}
  
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
+
4) if $m\leq f\leq M$ and $f$ is measurable, then $f\in L_1(X,\mu)$ 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/l05786053.png" /></td> </tr></table>
+
\begin{equation}m\mu X\leq\int\limits_X f\ d\mu\leq M\mu X\end{equation}
  
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
+
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
  
<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>
+
\begin{equation}\lim\limits_{n\to\infty}\int\limits_{E_n} f\ du\end{equation}
  
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
+
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.
  
<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>
+
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/l05786063.png" /> is measurable for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786064.png" />, if
+
If $A$ is a
 +
[[Measurable set|measurable set]] in $X$, 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/l05786065.png" /></td> </tr></table>
+
\begin{equation}\int\limits_A f\ d\mu\end{equation}
  
and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786066.png" />, then
+
is defined either as above, by replacing $X$ by $A$, 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/l05786067.png" /></td> </tr></table>
+
\begin{equation}\int\limits_X f\chi_A\ d\mu\end{equation}
  
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
+
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 $A_1\subset A$.
  
<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>
+
If
  
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).
+
\begin{equation}A=\bigcup_{n=1}^\infty A_n\end{equation}
  
The function of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057860/l05786074.png" /> given by
+
if $A$ is measurable for every $n$, 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/l05786075.png" /></td> </tr></table>
+
\begin{equation}A_n\cap A_k\ \text{for}\ n\neq k\end{equation}
  
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]].
+
and if $f\in L_1(A,\mu)$ then
  
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
+
\begin{equation}\int\limits_A f\ d\mu=\sum_{n=1}^\infty \int\limits_{A_n} f\ d\mu\end{equation}
  
<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>
+
Conversely, if under these conditions on $A_n$ one has $f\in L_1(A,\mu)$ for every $n$ and if $\sum_{n=1}^\infty\int\limits_{A_n} |f|\ d\mu < \infty$, then $f\in L_1(A,\mu)$ and the previous equality is true ($\sigma$-additivity of the Lebesgue integral).
  
For other measures this functional is called a [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes 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]].
  
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
+
For functions $f : \RR^n \to \RR^1$ the name "Lebesgue integral" is applied to the corresponding functional if the measure $\mu$ is the
 +
[[Lebesgue measure|Lebesgue measure]]; here, the set of summable functions is denoted simply by $L_1(\RR^n)$, 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/l05786087.png" /></td> </tr></table>
+
$$\int_{\RR^n} f(x) dx.$$
 +
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/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
+
If $f : [a, b] \to \RR^1$, $f \in L_1[a, b]$ and if $x : [\alpha, \beta] \to [a, b]$ 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/l05786094.png" /></td> </tr></table>
+
$$\int_a^b f(x) dx = \int_\alpha^\beta f(x(t)) x'(t) dt.$$
 +
If $f: [a, b] \to \RR^1$, $f \in L_1[a, b]$ and if $g:[a, b] \to \RR^1$ is monotone on $[a, b]$, then $fg \in L_1[a, b]$ and there is a point $\xi \in [a, b]$ such that
  
 +
$$\int_a^b f(x) g(x) dx = g(a) \int_a^\xi f(x) dx + g(b) \int_\xi^b f(x) dx$$
 
(the second mean-value theorem).
 
(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.
+
In 1902 H. Lebesgue gave (see {{Cite|Le}}) a definition of the integral for $X \subset \RR$ and measure $\mu$ equal to the Lebesgue measure. He constructed simple functions that uniformly approximate almost-everywhere on a set $f : E \to \RR^1$ of finite measure a measurable non-negative function $f$, and proved the existence of a common limit (finite or infinite) of the integrals of these simple functions as they tend to $f$. 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 $f: \RR^1 \to \RR^1$ from all possible generalized integrals.
  
 
====References====
 
====References====
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====Comments====
 
====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).
+
For other generalizations of the notion of an integral see
 +
[[A-integral|$A$-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====
 
====References====
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|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|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|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|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}}

Latest revision as of 05:26, 8 August 2018

2020 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 \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) 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.

If $A$ is a 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 $\sum_{n=1}^\infty\int\limits_{A_n} |f|\ d\mu < \infty$, 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); 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.

For functions $f : \RR^n \to \RR^1$ the name "Lebesgue integral" is applied to the corresponding functional if the measure $\mu$ is the Lebesgue measure; here, the set of summable functions is denoted simply by $L_1(\RR^n)$, and the integral by

$$\int_{\RR^n} f(x) dx.$$ For other measures this functional is called a Lebesgue–Stieltjes integral.

If $f : [a, b] \to \RR^1$, $f \in L_1[a, b]$ and if $x : [\alpha, \beta] \to [a, b]$ is a non-decreasing absolutely continuous function, then

$$\int_a^b f(x) dx = \int_\alpha^\beta f(x(t)) x'(t) dt.$$ If $f: [a, b] \to \RR^1$, $f \in L_1[a, b]$ and if $g:[a, b] \to \RR^1$ is monotone on $[a, b]$, then $fg \in L_1[a, b]$ and there is a point $\xi \in [a, b]$ such that

$$\int_a^b f(x) g(x) dx = g(a) \int_a^\xi f(x) dx + g(b) \int_\xi^b f(x) dx$$ (the second mean-value theorem).

In 1902 H. Lebesgue gave (see [Le]) a definition of the integral for $X \subset \RR$ and measure $\mu$ equal to the Lebesgue measure. He constructed simple functions that uniformly approximate almost-everywhere on a set $f : E \to \RR^1$ of finite measure a measurable non-negative function $f$, and proved the existence of a common limit (finite or infinite) of the integrals of these simple functions as they tend to $f$. 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 $f: \RR^1 \to \RR^1$ 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

Comments

For other generalizations of the notion of an integral see $A$-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

[H] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[P] I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian) MR0264015 Zbl 0206.06401
[S] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[Ro] H.L. Royden,

"Real analysis", Macmillan (1968)

[Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24 MR1736644 MR1645547 MR0924157 MR0850722 MR0662565 MR0344043 MR0210528 Zbl 1038.00002 Zbl 0954.26001 Zbl 0925.00005 Zbl 0613.26001 Zbl 0925.00003 Zbl 0278.26001 Zbl 0142.01701
[HS] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
How to Cite This Entry:
Lebesgue integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_integral&oldid=29352
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article