This is a problem of Lebesgue measure and measure theory specifically.
Suppose that
$f:\mathbb{R}^2\longrightarrow [0,\infty)$ is measurable.
$\Omega_1\subseteq \mathbb{R}^2$ is Lebesgue measurable, and,
$\Omega=\{(x,y,z)\in \mathbb{R}^3|(x,y)\in \Omega_1, 0\leq z\leq f(x,y)\}$
Show that $\Omega$ is Lebesgue measurable in $\mathbb{R}^3$ and that $|\Omega|_3=\int_{\Omega_1}fdxdy$
I don't know how to begin to solve it.