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.

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Can you do the case where $f = 1_A$ for some measurable $A \subset \mathbb{R}^2$? – Nate Eldredge Nov 15 '12 at 2:02
From here you can get some inspiration. – leo Nov 15 '12 at 6:18