I am so confusing that represent the Riemann Sum in terms of the Simple function. Consider initial tagged partition approach to Riemann Sum is $$\sum\limits_{i=0}^n f(t_i)(x_{i+1}-x_i)$$ This representation makes sense to me, however, I also have seen the representation(simple function) like below$$\sum\limits_{i=0}^nf(t_i)\chi_{[x_i,x_{i+1}]}$$.
Now consider a constant function, $f:\mathbb{R}\to \{c\}$, Riemann sum approach I could see it's calculating the area under the line $c$, but that characteristic functions taking values $\{0,1\}$ makes me hard to see what is going on.
Please enlighten me. Appreciate for any help.
It's the same approche. Notice that if $$g(x)=\sum_{i=1}^n f(t_i)\chi_{[x_i,x_{i+1}]},$$ then you define $$\int g:=\sum_{i=1}^n f(t_i)(x_{i+1}-x_i).$$