I'm working on a question, stated as follows:
We say that a sequence $\{a_n\}_{n=1}^{\infty}$ in $[0,1]$ is equidistributed (in $[0,1]$) if and only if for all intervals $[c,d]\subset [0,1]$, $$ \lim_{n\to\infty}\frac{|\{a_1,\ldots ,a_n\}\cap [c,d]|}{n}=d-c. $$ (Here $|A|$ denotes the number of elements in the set $A$.) Let $\mu_N=\tfrac{1}{N}\sum_{1\leq n\leq N}\delta_{a_n}$ with $\delta_{a_n}$ the point mass measure at $a_n$, that is, for any subset $S\in [0,1]$, $$ \delta_{a_n}(S) = \begin{cases} 1 & \text{if }a_n\in S\\ 0 & \text{if }a_n\not\in S \end{cases}. $$ Show that $\{a_n\}\subset [0,1]$ is equidistributed if and only if $$ \lim_{N\to\infty}\int_{0}^{1}fd\mu_N=\int_{0}^{1}fd m, $$ for all continuous functions on $[0,1]$, where $m$ is Lebesgue measure.
I'll spare you my "proof", but I will outline my idea.
This is pretty obvious for characteristic functions of closed intervals, and a little algebraic manipulation shows that this holds for characteristic functions of open intervals -- and hence characteristic functions of disjoint unions of open intervals -- as well. Every open set (on the real line) can be written as a disjoint union of open intervals, so it holds for characteristic functions of open sets. Every measurable set is $G_{\delta}$ set mod a null set, and from this it's not too hard to show that the result holds for characteristic functions of measurable sets. By linearity of limits and integration, we have the result for simple functions.
Let $f\in C[0,1]$ and $\{\varphi_n\}_{n=1}^{\infty}$ be a sequence of simple functions increasing pointwise to $f$. Then (by the monotone convergence theorem) $$ \int f dm =\lim_{n\to\infty}\int\varphi_n dm=\lim_{n\to\infty}\lim_{N\to\infty}\int_{0}^{1}\varphi_nd\mu_N. $$ Right here I want switch the order of limits, but I just can't justify doing it to myself.
Am I going about this problem the wrong way? I seems that approaching the problem this way not only proves it for continuous functions, but all measurable functions on $[0,1]$, which makes me a little suspicious. I figured since it was so obvious for characteristic functions of closed intervals, this was the natural approach, but I could (as I often am) be wrong. Any ideas or insights are greatly appreciated.