I am following this Wikipedia page on naive Monte Carlo method. Here is my slightly modified "proof"/reasoning for the $\mathcal{O} (N^{-1/2})$ speed of convergence of the method. I would like to have comments on whether I have understood the reasoning correctly.
The method: Let $\Omega \subset \mathbb{R}^n $ and let $\mu$ denote the $n$-dimensional Lebesgue measure. Let $x_1, x_2, \dots, x_N$ be a sample from the uniform distribution of $\Omega$. Thus, the probability density function is
$$ \rho(x)=\begin{cases} 1/\mu\left(\Omega\right){,}&\mathrm{in}\ \Omega{,}\\ 0{,}&\mathrm{otherwise.} \end{cases}$$
Then the method $$ \int_\Omega f \mathrm{d}\mu \simeq \frac{\sum_{k=1}^N f(x_i)}{N} \mu(\Omega)$$ has $\mathcal{O} (N^{-1/2}) $ speed of convergence.
Proof: Let $X_1, X_2, \dots, X_N$ be i.i.d. random variables. The distribution is the uniform distribution on $\Omega$. Thus, $f(X_1), f(X_2), \dots, f(X_N)$ are i.i.d. random variables as well. We denote the variance of $f(X_1), f(X_2), \dots, f(X_N)$ by $\sigma^2$.
The speed of convergence, i.e., the standard deviation of the method is
$$ \sqrt{\mathrm{Var}\left(\frac{\sum_{k=1}^Nf\left({X}_k\right)}{N}\mu\left(\Omega\right)\right)} = \sqrt{\frac{\mu \left(\Omega \right)^2}{N^2}\mathrm{Var}\left(\sum _{k=1}^Nf\left({X}_k\right)\right)} \\ = \sqrt{\frac{\mu \left(\Omega \right)^2}{N^2}\sum _{k=1}^N\mathrm{Var}\left(f\left({X}_k\right)\right)} \\ = \sqrt{\frac{\mu\left(\Omega\right)^2}{N^2}N\sigma^2} =\frac{\mu \left(\Omega \right)\sigma }{\sqrt{N}}. $$ Here $\mu(\Omega) \sigma $ is constant. Thus, the standard deviation is $\mathcal{O} (N^{-1/2})$.
By saying that the method converges, we mean that for large $N$, the standard deviation is close to zero. That is, for large enough $N$, "on average" it holds that $$ \left(\frac{\sum_{k=1}^N f(X_i)}{N} \mu(\Omega)-\int_\Omega f \mathrm{d}\mu \right)^2 \approx 0,$$ since $$ \mathbb{E} \left(f(X_i)\right)=\frac{1}{\mu(\Omega)} \int_{\Omega} f \mathrm{d}\mu. $$
However, that does not guarantee that the result or even the average of results of the method are close to 0. By law of large numbers, the average is close to zero with a high probability.