Full question: For a bounded function $f:[\mathbf{a,b}] \to \mathbb{R}$, set
$\displaystyle i(f) = \inf \left\{\int_a^b g\, d\lambda: g \in C[\mathbf{a,b}], g \geq f\right\}$, $\displaystyle s(f) = \sup \left\{\int_a^b h \,d \lambda : h \in C [\mathbf{a,b}], h \leq f \right\}$.
Show that $i(f) = \int_a^b f \,d\lambda$ and $s(f) = \int_a^b f \,d\lambda$
I'm pretty sure that I need to use that $d\mu = F' d\lambda$, but I'm not really sure where to go from there.