Lebesgue integral
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 |
Lebesgue integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_integral&oldid=43405