Riemann integral

Def Def 28.1

$$R(f,\xi )=\sum _j^{||\xi ||}{\Delta }_{j}f({\xi }_j)$$

$\sup R_{\sigma }(f,\xi )={\overline{\sum }}_{\sigma }f$
Where $\overline{\sum }$ is in Darboux upper sum

Def 30.1

$$\underset{n⟶\infty }{\lim }\frac{1}{n}\sum _{j=1}^{n}f(a+j\frac{b-a}{n})={\int }_a^{b}f(x)dx$$

Or
${\lim }_{n\to \infty }{\sum }_{i=0}^{n-1}f(x_i)(x_{i+1}-x_i)={\int }_a^{b}f(x)dx$
Basically we just need the parition be small enough.

Lemma 30.2

1. $R(f+\lambda g,\xi )=R(f,\xi )+\lambda R(g,\xi )$
2. for $[a,b]=[a,c]\cup [c,b]$
   1. $\sigma ={\sigma }_{[a,c]}\cup {\sigma }_{[c,d]}$ where it’s partition
   2. ${\Delta }_{{\sigma }_{[a,c]\cup \{c\}}}+{\Delta }_{{\sigma }_{[c,b]\cup \{c\}}}\le 2{\Delta }_{\sigma }$

Def 10.2

$${\Delta }_{\sigma }(f)=\sum (x_{j+1}-x_j)(\sup f-\inf f)$$