Recently I have been researching about equidistributed sequences. In the Wikipedia page: https://en.wikipedia.org/wiki/Equidistributed_sequence, I found the next theorem:
The sequence $\{s_n\}_{n=1}^\infty$ is equidistributed on $[a,b]$ if and only if for every Riemann-integrable function $f:[a,b]\to\mathbb{R}$ , the following limit holds: $$\lim\limits _{N\to\infty}\frac{1}{N}\sum\limits _{n=1}^{N}f(s_{n})=\int_{a}^{b}f(x)\text{ d}x$$
(In the original version $f$ is complex-valued, but this version is very easily verified as true as well).
Furthermore, in the same Wikipedia page it says:
de Bruijn–Post Theorem states the converse of the above criterion: If $f$ is a function such that the criterion above holds for any equidistributed sequence in $[a,b]$, then $f$ is Riemann-integrable in $[a,b]$.
I was not successful in finding any proof for that last theorem, not by myself nor online. My questions are:
- Can you present a proof of de Bruijn–Post Theorem, or maybe give me a link to a useful resource?
- Would it be possible to redefine Riemann-integrable function as one that satisfies that criterion, and then to construct the entire theory from this direction?