Riemann可積分

$ \exist A\in\R\forall\varepsilon>0\exist\delta>0\forall x:[0,N]\cap\Z\to[a,b]\forall x^*:]0,N]\cap\Z\to[a,b]:

$ \begin{dcases}a= x_0\land\forall i\in\N_{\le N}: x_{i-1}< x_i\land x_N=b\\\forall i\in\N_{\le N}: x_{i-1}\le x_i^*\le x_i\\\max_{0<i\le N}\varDelta x_i<\delta\end{dcases}\implies\left|\sum_{0<i\le N}f( x^*_i)\varDelta x_i-A\right|<\varepsilon

もう少し整理する

proof:

$ S(f,x_\bullet,t_\bullet)=\sum_{0<i\le N}f( t_i)\varDelta x_i

$ =c\sum_{0<i\le N}\varDelta x_i

$ =c(x_N-x_0)

$ =c(\max I-\min I)

$ \implies\forall\varepsilon>0\exist I\in\mathcal P(I)\exist\forall t_\bullet:x_\bullet\text{の代表点列}:|S(f,x_\bullet,t_\bullet)-c(\max I-\min I)|=0<\varepsilon

$ \underline{\implies\int_Ic\mathrm dx=c(\max I-\min I)\quad}_\blacksquare

proof:

$ \iff\forall\varepsilon>0\exist\delta>0\forall x,y\in I:|x-y|<\delta\implies|f(x)-f(y)|<\varepsilon

proof:

proof

$ \top

$ \implies\forall I\subseteq[0,1]:\begin{dcases}I\cap\Bbb{Q}\neq0\\ I\cap(\R\setminus\Bbb{Q})\neq0\end{dcases}

$ \implies\forall x\in\mathcal P([0,1])\forall i\in\N_{\le N}:\begin{dcases}\sup f^\to([x_i,x_{i-1}])=1\\\inf f^\to([x_i,x_{i-1}])=0\end{dcases}

$ \implies\begin{dcases}\overline{\int_0^1}f(x)\mathrm dx=\inf_{x\in\mathcal P([0,1])}\sum_{0<i\le N}\sup f^\to([x_i,x_{i-1}])\varDelta x_i=\inf_{x\in\mathcal P([0,1])}(1-0)=1\\\underline{\int_0^1}f(x)\mathrm dx=\sup_{x\in\mathcal P([0,1])}\sum_{0<i\le N}\inf f^\to([x_i,x_{i-1}])\varDelta x_i=\sup_{x\in\mathcal P([0,1])}\cdot0=0\end{dcases}

$ \implies\overline{\int_0^1}f(x)\mathrm dx\neq\underline{\int_0^1}f(x)\mathrm dx

性質

$ \int_I F'(x)\mathrm dx=F(\max I)-F(\min I)

References