for a test prep exam
Suppose that $f: R^2 \to [0, \infty)$ is measurable, $\Omega_1 \subseteq R^2$ is Lebesque measurable and $\Omega = \{(x, y, z) \in R^3 | (x, y) \in \Omega_1, 0 \leq z \leq f(x, y)\}$. Show that $\Omega$ is Lebesgue measurable in $R^3$ and $|\Omega|_3 = \int_{\Omega_1} fdxdy$
What i tried:
I tried to cover the set $\Omega$ with an open set, to verify the definition of lebesgue-measurable...I tried to use the measure of the function $f$ but I got nothing, then after a few hours I stuck