Let $\mathcal{X} \subset \mathbb{R}^d$ be a compact domain and $\mu$ be a finite measure defined on $\mathcal{X}$. For simplicity, we can assume that $\mathcal{X} = [0,1]^d$ and $\mu$ is the Lebesgue measure. Let $f : \mathcal{X} \to \mathbb{R}$ be a function such that $f \in L^2(\mu)$. Further, let $\{x_i\}_{i = 1}^{n}$ be a set of $n$ i.i.d. samples drawn from to $\mu$. I am interested in bounding the error given by: $$ \left| \frac{1}{n} \sum_{i = 1}^{n} f^2(x_i) - \int_{\mathcal{X}} f^2 d\mu \right|. $$
Intuitively, this error should be decreasing with $n$ similar to the case of Riemann sums or other concentration inequalities. I was wondering if there is a result that bound this error in terms of $n$? Any leads or references are appreciated.