1 设 $$ex phi(x)=sum_{i=1}^j c_ichi_{E_i}(x),quad c_igeq 0, eex$$
其中 $$ex E_imbox{ 可测},quad E_imbox{ 两两不交},quad E=cup_{i=1}^j E_i, eex$$
则定义 $$ex int_E phi(x) d x=sum_{i=1}^j c_icdot mE_i. eex$$
若 $A(subset E)$ 可测, 则定义 $$ex int_Aphi(x) d x=sum_{i=1}^j c_icdot m(E_icap A). eex$$
2 例: $dps{D(x)=sedd{a{ll} 1,&xinbQ,\ 0,&xinbRs bQ ea}}$ 的积分为 $$ex int_{bR}D(x) d x =1cdot m(bQ)+0cdot m(bRs bQ)=0. eex$$
3 性质: 设 $phi(x),psi(x)$ 为非负简单函数, 则
(1) 正齐次性 $$ex cgeq 0 a int_Ecphi(x) d x =cint_E phi(x) d x. eex$$
证明: $$eex ea int_Ecphi(x) d x =sum_{i=1}^j cc_icdot mE_i =csum_{i=1}^j c_icdot mE_i =cint_Ephi(x) d x. eea eeex$$
(2) 有限可加性 $$ex int_E[phi(x)+psi(x)] d x =int_E phi(x) d x +int_E psi(x) d x. eex$$
证明: $$eex ea &quad phi(x)=sum_{i=1}^j c_ichi_{E_i},quad psi(x)=sum_{k=1}^l d_kchi_{F_k}\ & a phi(x)+psi(x) =sum_{i=1}^j sum_{k=1}^l (c_i+d_k)chi_{E_icap F_k}\ & a int_E[phi(x)+psi(x)] d x =sum_{i=1}^j sum_{k=1}^l (c_i+d_k)cdot m(E_icap F_k)\ &qquadqquadqquad = sum_{i=1}^j c_isum_{k=1}^l m(E_icap F_k) +sum_{k=1}^l d_ksum_{i=1}^jm(E_icap F_k)\ &qquadqquadqquad =sum_{i=1}^j c_icdot mE_i +sum_{k=1}^l d_kcdot mF_k\ &qquadqquadqquad = int_Ephi(x) d x +int_Epsi(x) d x. eea eeex$$
(3) 对积分区域的有限可加性 $$ex A,B(subset E)mbox{ 可测} a int_{Acup B}phi(x) d x =int_Aphi(x) d x +int_Bphi(x) d x. eex$$
证明: $$eex ea int_{Acup B}phi(x) d x &=sum_{i=1}^j c_icdot m(Ecap(Acup B))\ &=sum_{i=1}^j c_i cdot [m(Ecap A)+m(Ecap B)]\ &quadsex{mbox{在可测集 }Ambox{ 的定义中取试验集 }T=Ecap (Acap B)}\ &=int_Aphi(x) d x +int_Bphi(x) d x. eea eeex$$
(4) 单增积分区域的极限 $$ex A_i(subset E)mbox{ 单增} a lim_{i oinfty}int_{A_i}phi(x) d x =int_{lim_{i oinfty}A_i}phi(x) d x. eex$$
证明: $$eex ea lim_{i oinfty}int_{A_i}phi(x) d x &=lim_{i oinfty}sum_{i=1}^j c_icdot m(Ecap A_i)\ &=sum_{i=1}^jc_icdot m sex{Ecap lim_{i oinfty}A_i}\ &=int_{lim_{i oinfty}A_i}phi(x) d x. eea eeex$$
4 作业: Page 132 T 2.