Given N uniform samples
$$ \overline{x}_1{,}\ \overline{x}_2{,}\ \dots{,}\ \overline{x}_N \in \Omega ,$$ I am trying to prove that Monte Carlo integration works, so $$ \int_{\Omega}^{ }f\left(\overline{x}\right)d\overline{x}=\int_{\Omega}^{ }d\overline{x}\cdot\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^Nf\left(\overline{x}_i\right).$$
So far I have got (by law of large numbers) that $$ \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^Nf\left(\overline{x}_i\right)=\mathbb{E}\left(f\left(\overline{x}\right)\right)=\int_{\Omega}^{ }f\left(\overline{x}\right)\rho\left(f\left(\overline{x}\right)\right)d\overline{x},$$ where $ \rho\left(f\left(\overline{x}\right)\right) $ is probability density function of $ f \left( \overline{x}\right).$
So is it automatically true that $ \rho \left( f \left( \overline{x} \right) \right) $ is the same as probability density function of $ \overline{x}?$ Or how could the proof be completed? Since $ \overline{x} $ is uniformly distributed, does it follow that $ f \left( \overline{x} \right) $ is uniformly distributed, too?