Exercise 12, § 1

One says that a sequence $(x_n)$ of real numbers in the interval $[0,\ 1]$ is uniformly distributed in this interval if

for every pair of numbers $\alpha,\ \beta$ such that $0 \leq \alpha \leq \beta \leq 1$, where $\nu_n(\alpha, \beta)$ denotes the number of indices $i$ such that $1 \leq i \leq n$ and $\alpha \leq x_i \leq \beta$.

Show that if the sequence $(x_n)$ is uniformly distributed and if $\mathbf{f}$ is a regulated function on $[0,\ 1]$, one has

(reduce to the case where $\mathbf{f}$ is a step function). Converse.

Show that for the sequence $(x_n)$ to be uniformly distributed it is enough that the relation (2) should hold for every real function $f$ belonging to a dense set in the space of real continuous functions on $[0,\ 1]$ endowed with the topology of uniform convergence.