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.]



Criterion for convergence of sums for non-continuous functions
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. 


