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 \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.