Criterion for convergence of sums for non-continuous functions

Question

The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{x_n\}$ of points in $G$ is equidistributed if for all $f\in C(G)$, we have equality: $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N f(x_n) = \int_G f(x)\, \mathrm{d}x . $$ Here, we fix the sequence $\{x_n\}$ and vary $f\in C(G)$. My question is based on what happens when we fix $f$ and vary the $\{x_n\}$. More precisely:

Let $f\in L^1(G)$. Call $f$ equidistribution-good if the equality above holds for all equidistributed $\{x_n\}$, even though $f$ is not necessarily continuous. Is there a good analytic condition (weaker than continuity) such that if $f$ satisfies this condition, then $f$ is equidistribution-good?

If $G=S^1$, then bounded variation works, though it's not clear to me that $f$ is equidistribution-good if and only if it has bounded variation (I think bounded + a.e.-continuity is iff here).

Any partial answers or pointers to references would be super helpful!

Even better, it would be awesome if there was a good analytic criterion for which functions $f$ aren't just equidistribution-good, but satisfy $$ \left|\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N f(x_n) - \int_G f(x)\, \mathrm{d}x \right| = O_{\{x_n\},f}(\text{some kind of ``discrepancy''}) . $$

[EDIT: I've modified the question to reflect Christian's comments.]

Answer 1

The paper "λ-equidistributed sequences of partitions and a theorem of the De Bruijn–Post type", by Chersi and Volčič, proves that if $(X,d,\lambda)$ is a separable metric space with probability measure whose support is $X$, then "$\frac{1}{N}\sum_{n\leqslant N} f(x_n)\to \lambda(f)$ for all $\lambda$-equidistributed $\{x_n\}$" if and only if $f$ is continuous almost everywhere.