# 勒贝格控制收敛定理

## 叙述

${\displaystyle (S,\Sigma ,\mu )}$为一个测度空间${\displaystyle (f_{n})_{n\geq 0}}$是一个实值的可测函数列。如果${\displaystyle (f_{n})}$逐点收敛于一个函数${\displaystyle f}$，并存在一个勒贝格可积的函数${\displaystyle g\in L^{1}}$，使得对每个${\displaystyle n\geq 0}$，任意${\displaystyle x\in S}$，都有

${\displaystyle |f_{n}(x)|\leq g(x)}$

1. ${\displaystyle f}$也是勒贝格可积的，${\displaystyle f\in L^{1}}$
2. ${\displaystyle \int _{S}fd\mu =\int _{S}\lim _{n\to \infty }f_{n}\,d\mu =\lim _{n\to \infty }\int _{S}f_{n}\,d\mu .}$

## 证明

${\displaystyle \forall x\in S\ |f(x)|\leq g(x)}$（于是${\displaystyle \scriptstyle f\in L^{1}}$）。

${\displaystyle |f-f_{n}|\leq 2g}$ 以及
${\displaystyle \limsup _{n\to \infty }|f-f_{n}|=0.}$

${\displaystyle \limsup _{n\to \infty }\int _{S}|f-f_{n}|\,d\mu \leq \int _{S}\limsup _{n\to \infty }|f-f_{n}|\,d\mu =0.}$

${\displaystyle {\biggl |}\int _{S}f\,d\mu -\int _{S}f_{n}\,d\mu {\biggr |}={\biggl |}\int _{S}(f-f_{n})\,d\mu {\biggr |}\leq \int _{S}|f-f_{n}|\,d\mu ,}$

## 控制函数的必要性

${\displaystyle \int _{0}^{1}\lim _{n\to \infty }f_{n}(x)\,dx=0\neq 1=\lim _{n\to \infty }\int _{0}^{1}f_{n}(x)\,dx,}$

${\displaystyle \int _{0}^{1}h(x)\,dx\geq \int _{1/m}^{1}h(x)\,dx=\sum _{n=1}^{m-1}\int _{\left({\frac {1}{n+1}},{\frac {1}{n}}\right]}n\,dx=\sum _{n=1}^{m-1}{\frac {1}{n+1}}\to \infty \quad }$ （当 ${\displaystyle m\to \infty }$ 时）

## 参考资料

• R.G. Bartle, "The Elements of Integration and Lebesgue Measure", Wiley Interscience, 1995.
• H.L. Royden, "Real Analysis", Prentice Hall, 1988.
• D. Williams, "Probability with Martingales", Cambridge University Press, 1991, ISBN 0-521-40605-6