Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is the set $S$ measurable (i.e. can we assign an area to it)? And if so, then how to compute the area of $S$?
I am looking for an argument in terms of the notions of step functions and step regions.
Since $f$ is a non-negative function defined, bounded, and integrable on the closed interval $[a,b]$, then the area of $S=\{(x,y) \mid x\in [a,b], y\in [0, f(x)]\}$ is the double integral:
$\begin{align}|S| & = \int_a^b \int_0^{f(x)}1 \;\mathrm{d}y\;\mathrm{d}x \\ & = \int_a^b [y]_{y=0}^{y=f(x)} \;\mathrm{d}x \\ & = \int_a^b f(x) \;\mathrm{d}x \end{align}$