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定义：设\(\,[a,\,b]\subset\mathbb{R},\)称\(\,\mathfrak{P}([a,b])=\{P\mid a,b\in P\subset[a,b],|P|< \infty\}\).+2
\(\qquad\)的成员\(\,P=\{x_0,x_1,\ldots,x_n\}\;(a=x_0\le x_{k-1}< x_k< x_n=b,\);gQB.&
\(\qquad{\small 0<\,}k< n\small=|P|)\)为\([a,b]\)的一个划分.记\(\,\Delta_k=x_k-x_{k-1}\scriptsize\;(k=\overline{1,n}).\)Wr
\(\qquad\)设\(\,f:[a,b]\to\small\mathbb{R}\,\)有界,记\(\,m_k(M_k)=\inf(\sup) f([x_{k-1},x_k]),\)-/0LO
\(\qquad\)称\(L(f,P)\displaystyle{\small\,=\sum_{k=1}^n} m_k\Delta_k,\,U(f,P){\small\,=\sum_{k=1}^n}M_k\Delta_k.\)为\(\,f\)关于划分\(\,P\,\)fF4
\(\qquad\)的下(上)和,\(\;L(f,[a,b])=\sup\,L(f,\mathfrak{P}([a,b])),\quad U(f,[a,b])\)n|R
\(\qquad=\inf\,U(f,\mathfrak{P}([a,b]))\,\)分别称为\(\,f\,\)在\([a,b]\)上的下(上)积分.N7zcm#
\(\qquad\)模与振幅:\(\,\left\|P\right\|=\max\{\Delta_k:1\le k\le|P|\},\;\;\omega_k=M_k-m_k\)$b2
定理：\(\small\,P,(\subset)P^*\in\mathfrak{P}([a,b])\implies L(f,P)\le L(f,P^*)\le U(f,P^*)\le U(f,P)\)W
证：易证当\(\,P^*\)比\(\underset{\,}{\,}P\)多一点时结论成立,据归纳法,一般结论成立.lS
定理：\(\,\forall P_1,\,P_2\in\mathfrak{P}([a,b]),\;L(f,P_1)\le U(f,P_2)\).z6@
证：令\(\,P=P_1\cup P_2(\in\mathfrak{P}([a,b])),\)由上定理易见有以下不等式:NQ
\((^*)\quad\quad\qquad\,L(f,P_1)\le L(f,P)\le U(f,P)\le U(f,P_2)\underset{\,}{.}\)Et<Dk2
推论：\(\,\displaystyle\underset{^{-\;\;}}{\int_a}^b f:=L(f,[a,b])\le U(f,[a,b])=:\bar{\int_a^b} f\)[QH\
定义：若\(\,L(f,[a,b])=U(f,[a,b])=\lambda\in\mathbb{R},\,\)则称\(\,f(\in\mathscr{R}([a,b]))\,\)黎*b:
\(\qquad\)曼可积.\(f\,\)在\([a,b]\)上的积分\(\displaystyle\,\int_a^b f=\lambda.\)((nW
定理：\(\,f\in\mathscr{R}([a,b])\iff\forall\varepsilon>0\,\exists P\in\mathfrak{P}:\;U(f,P)-L(f,P)<\varepsilon\)U%l
证：\((\hspace{-3px}\implies\hspace{-4px}):\,\forall\varepsilon>0\,\exists\,P_1,P_2\in\mathfrak{P}([a,b]):\)IBM
\(\qquad 0< {\Large\int}_a^b f-L(f,P_1),\;\;U(f,P_2)-{\Large\int}_a^b f<\large\frac{\varepsilon}{2}.\) 取\(\,P=P_1\cup P_2,\)bMp
\(\qquad\)据\((^*),\;0<{\Large\int}_a^b f-L(f,P),\;\;U(f,P)-{\Large\int}_a^b f<\large\frac{\varepsilon}{2}.\;\)二不等式相b{v2P
\(\qquad\)加即得\(\,U(f,P)-L(f,P)< \varepsilon.\;P=P_1\cup P_2\in\mathfrak{P}([a,b]).\)WQJF&
\(\qquad(\hspace{-1px}\Longleftarrow\hspace{-1px}):\displaystyle\bar{\int_a^b}f-\underset{^{-\;\;}}{\int_a}^b f\le U(f,P)-L(f,P)\;\;(\forall P\in\mathfrak{P}([a,b])).\small\;\square\)*1}T
推论：\((1)\;\;f\in\mathscr{R}([a,b])\hspace{-3px}\iff\hspace{-5px}{\displaystyle\lim_{\left\|P\right\|\to 0}}\sum_{k=1}^{\small|P|}\omega_k\Delta_k=0\)'~)
\(\quad\qquad\,(2)\;\,f\in\mathscr{R}([a,b])\implies{\displaystyle\int_a^b f=\lim_{n\to\infty}{\scriptsize\frac{b-a}{n}}\sum_{k=1}^n}f(a+{\large\frac{k(b-a)}{n}})\)HPTM%
\(\quad\qquad\,(3)\;\,\mathscr{C}([a,b])\subset\mathscr{R}([a,b])\)\Q
证：若\(\,f\in\mathscr{R}([a,b]),\;\varepsilon>0,\,\)取划分\(P\,\)使\(\,(U-L)(f,P)<\varepsilon/2.\) 令tYiu
\(\qquad\delta_P=\min\{\Delta_k\mid 1\le k\le|P|\},\;\)则对任意\(\,Q\in\mathfrak{P}([a,b]),\,\)只要dRV
\(\qquad\left\|Q\right\|< \min(\delta_P,\frac{\varepsilon}{2|P|(\omega+1)})\),就有\(\small\,(U-L)(f,Q)\le(U-L)(f,Q\cup P)\)rl}b`.
\(\qquad+|P|\omega\left\|Q\right\|< {\small\dfrac{\varepsilon}{2}+\dfrac{\varepsilon}{2}}=\varepsilon.\;\therefore\;(1)\)的必要性得证.充分性证略.!krUg
\((2)\quad\)这是\((1)\)的直接推论,不难看出,\(a+\frac{k(b-a)}{n}\;\)可被任意a1)zN
\(\qquad\qquad\xi_k\in[a+\frac{(k-1)(b-a)}{n},a+\frac{k(b-a)}{n}]\) 取代.kZH
\((3)\quad\)若\(\,f\in\mathscr{C}([a,b]),\;\varepsilon>0,\;\)由\(\,f\,\)在\(\,[a,b]\)上的一致连续性,存在xSy
\(\qquad\;\delta>0\,\)使\(\,|f(s)-f(t)|< {\large\frac{\varepsilon}{b-a+1}}\;(\forall s,t\in[a,b],\,|s-t|\le\delta)\)CymOm
\(\qquad\,\)取\(\,P_n=\{a+k(b-a)/n\mid 0\le k\le n\},\;n>\lfloor\frac{b-a}{\delta}\rfloor\), 则易见%{p
\(\qquad\,U(f,P_n)-L(f,P_n)={\displaystyle\sum_{k=1}^n}\omega_k\Delta_k< {\large\frac{\varepsilon}{b-a+1}}(b-a+\frac{1}{2})< \varepsilon\)`
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2020/12/17 01:25pm 
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