1 设 $$\bex \phi(x)=\sum_{i=1}^j c_i\chi_{E_i}(x),\quad c_i\geq 0, \eex$$
其中 $$\bex E_i\mbox{ 可测},\quad E_i\mbox{ 两两不交},\quad E=\cup_{i=1}^j E_i, \eex$$
则定义 $$\bex \int_E \phi(x)\rd x=\sum_{i=1}^j c_i\cdot mE_i. \eex$$
若 $A(\subset E)$ 可测, 则定义 $$\bex \int_A\phi(x)\rd x=\sum_{i=1}^j c_i\cdot m(E_i\cap A). \eex$$
2 例: $\dps{D(x)=\sedd{\ba{ll} 1,&x\in\bbQ,\\ 0,&x\in\bbR\bs \bbQ \ea}}$ 的积分为 $$\bex \int_{\bbR}D(x)\rd x =1\cdot m(\bbQ)+0\cdot m(\bbR\bs \bbQ)=0. \eex$$
3 性质: 设 $\phi(x),\psi(x)$ 为非负简单函数, 则
(1) 正齐次性 $$\bex c\geq 0\ra \int_Ec\phi(x)\rd x =c\int_E \phi(x)\rd x. \eex$$
证明: $$\beex \bea \int_Ec\phi(x)\rd x =\sum_{i=1}^j cc_i\cdot mE_i =c\sum_{i=1}^j c_i\cdot mE_i =c\int_E\phi(x)\rd x. \eea \eeex$$
(2) 有限可加性 $$\bex \int_E[\phi(x)+\psi(x)]\rd x =\int_E \phi(x)\rd x +\int_E \psi(x)\rd x. \eex$$
证明: $$\beex \bea &\quad \phi(x)=\sum_{i=1}^j c_i\chi_{E_i},\quad \psi(x)=\sum_{k=1}^l d_k\chi_{F_k}\\ &\ra \phi(x)+\psi(x) =\sum_{i=1}^j \sum_{k=1}^l (c_i+d_k)\chi_{E_i\cap F_k}\\ &\ra \int_E[\phi(x)+\psi(x)]\rd x =\sum_{i=1}^j \sum_{k=1}^l (c_i+d_k)\cdot m(E_i\cap F_k)\\ &\qquad\qquad\qquad \ \ = \sum_{i=1}^j c_i\sum_{k=1}^l m(E_i\cap F_k) +\sum_{k=1}^l d_k\sum_{i=1}^jm(E_i\cap F_k)\\ &\qquad\qquad\qquad \ \ =\sum_{i=1}^j c_i\cdot mE_i +\sum_{k=1}^l d_k\cdot mF_k\\ &\qquad\qquad\qquad \ \ = \int_E\phi(x)\rd x +\int_E\psi(x)\rd x. \eea \eeex$$
(3) 对积分区域的有限可加性 $$\bex A,B(\subset E)\mbox{ 可测}\ra \int_{A\cup B}\phi(x)\rd x =\int_A\phi(x)\rd x +\int_B\phi(x)\rd x. \eex$$
证明: $$\beex \bea \int_{A\cup B}\phi(x)\rd x &=\sum_{i=1}^j c_i\cdot m(E\cap(A\cup B))\\ &=\sum_{i=1}^j c_i \cdot [m(E\cap A)+m(E\cap B)]\\ &\quad\sex{\mbox{在可测集 }A\mbox{ 的定义中取试验集 }T=E\cap (A\cap B)}\\ &=\int_A\phi(x)\rd x +\int_B\phi(x)\rd x. \eea \eeex$$
(4) 单增积分区域的极限 $$\bex A_i(\subset E)\mbox{ 单增}\ra \lim_{i\to\infty}\int_{A_i}\phi(x)\rd x =\int_{\lim_{i\to\infty}A_i}\phi(x)\rd x. \eex$$
证明: $$\beex \bea \lim_{i\to\infty}\int_{A_i}\phi(x)\rd x &=\lim_{i\to\infty}\sum_{i=1}^j c_i\cdot m(E\cap A_i)\\ &=\sum_{i=1}^jc_i\cdot m \sex{E\cap \lim_{i\to\infty}A_i}\\ &=\int_{\lim_{i\to\infty}A_i}\phi(x)\rd x. \eea \eeex$$
4 作业: Page 132 T 2.