Positivity of integral

Let $f: {\bf R}^n \rightarrow {\bf R}$ and

$\displaystyle\int_Q f(n) dV = \displaystyle\lim_{m \rightarrow \infty} \displaystyle\sum_i f( e_i ) \Delta V$, a $(n+1)$-dimensional volume,

where $\Delta V$ a $n$-dimensional volume of an $n$-dimensional box, is the Riemann integral.

The integral of a positive function is positive.

If $f(u) \geq 0$ for all $u \in Q$, then $\displaystyle\int_Q f(u) dV \geq 0$.

(Since $f(c_i) \geq 0, \Delta V \geq 0$ in Riemann sums)

It follows then, if $f(u) \geq g(u)$ for all $u ∈ Q$, then $\displaystyle\int_Q f(u) dV \geq \displaystyle\int_Q g(u) dV$.