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Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence which is equidistributed in $[0,1)$.

(a) Is it true that $$\frac{1}{N}\sum\limits_{n=1}^N f(x+\xi_n)$$ converges to the constant $\int_0^1 f(y) dy$ for each $x\in \mathbb{R}$ as $N\to \infty$ ?

(b) If so, is the convergence uniform over all $x$?

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    As you can approximate *Riemann* integrable functions from above and below by simple step functions, the answer has to be yes on both counts. It would be a different matter for Lebesgue integrable functions though.2010-12-27
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    @George. See my comments below.2010-12-27

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