I just read about the Lebesgue-Integral and wondered, why they defined it by taking advantage of these sequences of functions What we want is something like the space enclosed by the graph of the function and the Domain crossed 0, don't we? $$\int f d \mu_1 = \mu_2 (\{(x,y) \in D \times V )| 0\leq y \leq f(x)\}), f: D\rightarrow V, f\geq 0$$
$\mu_1$ is the Lebesgue Measure in $(\mathbb{R}^n,\Sigma_1, \mu_1)$, $\mu_2$ the Lebesgue-Measure in $(\mathbb{R}^{n+k},\Sigma_2,\mu_2), D \subseteq \mathbb{R}^{n}, V \subseteq \mathbb{R}^{k}$. So this only works if $V \subseteq \mathbb{R}^k$for a $n,k\in \mathbb{N}$.
I am pretty sure this is true, but why?