How prove this nice limit $\lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$ let $f(x)$ is Continuous on $[0,1]$, and such $f(0)=f(1)$,and if $a$ is irrational number.
show that

$$\lim_{n\to+\infty}\dfrac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$$

where $\{ka\}=x-[x]$,and $[x]$ is 
is the largest integer not greater than $x$
This problem  is from this ( problem 8)http://wenku.baidu.com/view/a643e6c26137ee06eff91855.html
and I find this Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
Have without Trigonometric series  methods? because  this problem is Freshman exam questions

There actually is a proof not using trigonometric series but heavily using ergodic theory. I guess that wouldn't be a freshman's solution.
Anyhow, the major idea behind the ergodic theory proof is as follows: One identifies $[0,1]$ with the unit circle $S^1 = \{e^{2\pi i t}\mid t \in \mathbb{R}\}$. Consider on $S^1$ the rotation $R$ with angle $2\pi\alpha$: $R(e^{2\pi i t}) = e^{2\pi i (t+\alpha)}$. The sequence $(\{k\alpha\})_k$ translates on $S^1$ to $(R^k(1))_k$. Now one endows $S^1$ with its Borel $\sigma$-algebra and shows that $R$ is uniquely ergodic with the Haar measure (corresponding to the Lebesgue measure on $[0,1]$) being the unique ergodic $R$-invariant measure. The statement then follows from Birkhoff's pointwise ergodic theorem. 
More details are contained in many textbooks on ergodic theory. This precise example is worked out in, e.g., Einsiedler, Ward: Ergodic Theory with a view towards Number Theory.
