Measurability of the region under a graph of a measurable function 
Let $f:{\bf R}^d\to[0,+\infty]$ be an unsigned measurable function. Show that the region 
  $\{(x,t)\in{\bf R}^d\times{\bf R}:0\leq t\leq f(x)\}$ is a Lebesgue measurable subset of ${\bf R}^{d+1}$.

For any unsigned simple function $\sum_{n=1}^Na_n1_{E_n}$ where $a_n\in[0,+\infty]$ and $E_n\subset{\bf R}^d$ are measurable and disjoint, this is true since  $\cup_{n=1}^NE_n\times[0,a_n]$ is measurable. How can I show the genral case?
 A: My previous answer was incorrect (thanks Jorge for pointing it out!). Here's a hopefully correct argument: Write $\Gamma(f) = \{(x,t) \in \Omega \times [0,+\infty] : 0 \leq t \leq f(x)\}$ for the region below the graph of a function $f \colon \Omega \to [0,+\infty]$. We want to show that $\Gamma(f)$ is measurable.
One can do this using a couple of reductions: 


*

*If $f \colon \Omega \to [0,+\infty)$ is a bounded measurable function on a bounded measurable set $\Omega$ we can find simple functions $g_n$ and $h_n$ such that $g_n \leq f \leq h_n$ and $h_n - g_n \leq \frac{1}{n}$. We know that $\Gamma(g_n)$ and $\Gamma(h_n)$ are measurable, that $\Gamma(g_n) \subseteq \Gamma(f) \subset \Gamma(h_n)$ and we have $\mu(\Gamma(h_n) \setminus \Gamma(g_n)) \leq \frac{1}{n} \mu(\Omega)$. This shows that $\Gamma(f)$ is measurable since it coincides with $\bigcup_n \Gamma(g_n)$ and $\bigcap_n \Gamma(h_n)$ up to a null set.

*If $f\colon \Omega \to [0,+\infty]$ is any measurable function on a bounded measurable set, approximate it by $f_n = \min\{f,n\}$ and observe that $\Gamma(f) = \bigcup_n \Gamma(f_n) \cup \left(\{x \in \Omega : f(x) = +\infty\} \times \{\infty\}\right)$. By the first step $\Gamma(f_n)$ is measurable, hence so is $\Gamma(f)$.

*If $\Omega \subseteq \mathbf{R}^n$ is an arbitrary measurable set, we can write $\Omega = \bigcup_{n} \Omega_n$ where $\Omega_n$ is bounded and measurable. Write $f_n = f|_{\Omega_n}$ and note that $\Gamma(f) = \bigcup_n \Gamma(f_n)$ is measurable because $\Gamma(f_n)$ is measurable by steps 1 and 2.
