For an article I am writing, I would like to know that two somewhat different looking conditions are in fact equivalent. Here is the setting. $X$ is a compact (and first countable) metric space and $\mu$ is a Radon probability measure on $X$. That is: $\mu$ is a measure on the $\sigma$-algebra of Borel serts of $X$ (the $\sigma$-algebra generated by the open sets), has total measure one, and is inner regular, that is the measure of any Borel set $B$ is the sup of the measures of the compact subsets of $B$.
Now let ${x_n}$ be a sequence in $X$. Let's say that this sequence is $\mu$-uniformly-distributed-A" "$\mu$-uniformly-distributed-A" if for any open subset $O$ of $X$ $$\mu(O) =\lim_{N \to \infty} { \#(O,N) \over N}$$ where $\#(O,N)$ is the number of $x_k$ in $O$ for $k = 1, \ldots, N$. (Or, in other words, the measure of $O$ is the "average number of the $x_n$ that are in $O$").