Equidistribution of a set of numbers $\{s_1, s_2, s_3, \cdots\}$, loosely speaking, is the property that the proportion of terms falling in a subinterval is proportional to the length of that subinterval (a more precise definition can be found in Wikipedia).
For simplicity let's restrict to the case of $s_n$ being real numbers in the interval $[0,1]$; then an example of an equidistributed sequence is $s_n = n\alpha \mod 1$ where $\alpha$ is an irrational number.
Is equidistribution equivalent to showing that the moments of the set of numbers match the moments of a uniform random variable $X$ on the interval, for all moments?
E.g. in the case of the interval $[0,1]$ can I say $\{s_n\}$ is equidistributed if $$ \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^n s_n^k = \mathbb{E}[X^k]=\int_0^1dx x^k=\frac{1}{k+1} $$ for all non-negative integer $k$?
Why or why not? Thanks.