Notation:
Riemann sum ${S_{p}}(f)$
Upper/Lower Riemann sum $\overline{S_{p}}(f)$/$\underline{S_{p}}(f)$
Upper/Lower Darboux Sum $U_p(f)$/$L_p(f)$
Definition:
Let $f:[a,b]\rightarrow \mathbb{R}$ and p be partition of [a,b]
Riemann sum:
$${S_{p}}(f)=\sum _{i=1}^n\:(x_i-x_{i-1})f(t_i)\text{ where } t_i\in[x_{i-1},x_i]$$
Upper/Lower Riemann sum:
$$\overline{S_{p}}(f)/\underline{S_{p}}(f)=\sup\{{S_{p}}(f)\}/\inf\{{S_{p}}(f)\}$$
Upper/Lower Darboux sum:
$$U_p(f)/L_p(f)=\sum _{i=1}^n\:(x_i-x_{i-1})\sup\limits _{x\in[x_{i-1},x_i]}f(x)/\sum _{i=1}^n\:(x_i-x_{i-1})\inf\limits _{x\in[x_{i-1},x_i]}f(x)$$
Question:
On my text book(maybe most text book), it only defined Riemann sum and Upper/Lower Darboux sum. And I also see some text use Upper/Lower Darboux sum to define Upper/Lower Riemann sum, are they equivalent definitions? In another word:
$$\sup\{{S_{p}}(f)\}/\inf\{{S_{p}}(f)\}\overset?=\sum _{i=1}^n\:(x_i-x_{i-1})\sup\limits _{x\in[x_{i-1},x_i]}f(x)/\sum _{i=1}^n\:(x_i-x_{i-1})\inf\limits _{x\in[x_{i-1},x_i]}f(x)$$
Any help would be appreciated.