# 可测函数

## 正式定义

${\displaystyle f^{-1}(B)\in \Sigma _{X}}$

## 重要范例

### 实可测函数

${\displaystyle {\mathcal {I}}={\bigg \{}A\in {\mathcal {P}}(\mathbb {R} )\,{\bigg |}\,(\exists a)(\exists b)\left[\,(a,\,b\in \mathbb {R} )\wedge (A=(a,\,b))\,\right]{\bigg \}}}$
${\displaystyle {\mathcal {B}}_{\mathbb {R} }:=\sigma ({\mathcal {I}})=\bigcap {\bigg \{}\Sigma \,{\bigg |}\,(\Sigma {\text{ is a sigma algebra.}})\wedge ({\mathcal {I}}\subseteq \Sigma ){\bigg \}}}$

### 博雷尔函数

${\displaystyle \sigma (\tau _{X})=\bigcap {\bigg \{}\Sigma \,{\bigg |}\,(\Sigma {\text{ is a sigma algebra.}})\wedge (\tau _{X}\subseteq \Sigma ){\bigg \}}}$
${\displaystyle \sigma (\tau _{Y})=\bigcap {\bigg \{}\Sigma \,{\bigg |}\,(\Sigma {\text{ is a sigma algebra.}})\wedge (\tau _{Y}\subseteq \Sigma ){\bigg \}}}$

## 可测函数的性质

${\displaystyle \Sigma =\left\{B\in {\mathcal {P}}(Y)\,{\big |}\,f^{-1}(B)\in \Sigma _{X}\right\}}$

${\displaystyle Y}$ σ代数

(1) ${\displaystyle Y\in \Sigma }$

${\displaystyle f^{-1}(Y)=\left\{x\in X\,|\,(\exists y\in Y)\left[f(x)=y\right]\right\}=X\in \Sigma _{X}}$

(2) ${\displaystyle B\in \Sigma }$  ，则 ${\displaystyle Y-B\in \Sigma }$

${\displaystyle B\in \Sigma }$  ，因为：

${\displaystyle f^{-1}(Y-B)={\big \{}x\in X\,|\,(\exists y)\left\{(y\in Y)\wedge (y\notin B)\wedge [f(x)=y]\right\}{\big \}}=X-f^{-1}(B)\in \Sigma _{X}}$

(3)可数个并集仍在 ${\displaystyle \Sigma }$

${\displaystyle \{B_{1},\,B_{2},\,\dots \}\subseteq \Sigma }$  ，那因为：

${\displaystyle f^{-1}\left(\bigcup \{B_{1},\,B_{2},\,\dots \}\right)={\big \{}x\in X\,{\big |}\,(\exists y)\left\{[f(x)=y]\wedge (\exists i\in N)(y\in B_{i})\right\}{\big \}}=\bigcup \{f^{-1}(B_{1}),\,f^{-1}(B_{2}),\,\dots \}\in \Sigma _{X}}$

1. 对所有 ${\displaystyle B\in {\mathcal {F}}_{Y}}$ ${\displaystyle f^{-1}(B)\in \Sigma _{X}}$
2. ${\displaystyle f}$ ${\displaystyle \Sigma _{X}}$  - ${\displaystyle \sigma ({\mathcal {F}}_{Y})}$  可测函数

(1 ${\displaystyle \Rightarrow }$  2)

${\displaystyle f^{-1}(B)\in \Sigma _{X}}$

${\displaystyle {\mathcal {F}}_{Y}\subseteq \left\{B\in {\mathcal {P}}(Y)\,{\big |}\,f^{-1}(B)\in \Sigma _{X}\right\}}$

${\displaystyle \sigma ({\mathcal {F}}_{Y})\subseteq \left\{B\in {\mathcal {P}}(Y)\,{\big |}\,f^{-1}(B)\in \Sigma _{X}\right\}}$

(2 ${\displaystyle \Rightarrow }$  1)

${\displaystyle {\mathcal {F}}_{Y}\subseteq \sigma ({\mathcal {F}}_{Y})\subseteq \left\{B\in {\mathcal {P}}(Y)\,{\big |}\,f^{-1}(B)\in \Sigma _{X}\right\}}$

• ${\displaystyle f:X\to Y}$ ${\displaystyle \Sigma _{X}}$ - ${\displaystyle \sigma (\tau _{Y})}$ 可测函数
• ${\displaystyle g:Y\to Z}$ ${\displaystyle \tau _{Y}}$  - ${\displaystyle \tau _{Z}}$  连续函数

“对所有的 ${\displaystyle C\in \tau _{Z}}$ ${\displaystyle {(g\circ f)}^{-1}(C)=f^{-1}[g^{-1}(C)]\in \Sigma _{X}}$

“对所有的 ${\displaystyle C\in \tau _{Z}}$ ${\displaystyle g^{-1}(C)\in \tau _{Y}\subseteq \sigma (\tau _{Y})}$

${\displaystyle f}$  又为 ${\displaystyle \Sigma _{X}}$ - ${\displaystyle \sigma (\tau _{Y})}$ 可测函数，故可以得到 ${\displaystyle f^{-1}[g^{-1}(C)]\in \Sigma _{X}}$  ，所以本定理得证。${\displaystyle \Box }$

• 两个可测的实函数的和与积也是可测的。
• 可数个实可测函数的最小上界也是可测的。
• 可测函数的逐点极限是可测的。（连续函数的对应命题需要比逐点收敛更强的条件，例如一致收敛。）
• 卢辛定理

## 勒贝格可测函数

${\displaystyle \{x\in \mathbb {R} :f(x)>a\}}$

## 参考文献

1. ^ Billingsley, Patrick. Probability and Measure. Wiley. 1995. ISBN 0-471-00710-2.