Exercise 34, § 1

For every compact commutative group G, denote by $\mu_G$ the normalized Haar measure on G.

a) Let G be a compact commutative group. Let $(\nu_n)_{n\in\mathbf{N}}$ be a sequence of measures on G such that $\nu_n(G) = 1$ for every $n$. Then $(\nu_n)$ converges vaguely to $\mu_G$ if and only if $\nu_n(\widehat{x})$ converges to 0 for every character $\widehat{x}\not= 1$ of G («Weyl equidistribution criterion»).

b) Let $r\geqslant 1$ be an integer and $x= (x_1, . . . , x_r)\in \mathbf{T}^r$. For $n\geqslant 0$, denote by $\nu_n$ the measure on $\mathbf{T}^r$ such that

$n-1$

1

$k=0$

for $f\in \mathscr{C}(\mathbf{T}^r)$. The sequence $(\nu_n)$ converges vaguely to the normalized Haar measure $\mu_H$, where H is the closure in $\mathbf{T}^r$ of the set $\{kx|k\in \mathbf{Z}\}$.

c) Let $y_i$ be real numbers such that $x_i$ is the class of $y_i$. If the family $(1, y_1, . . . , y_n)$ is $\mathbf{Q}$-linearly independent, then one has $H =\mathbf{T}^r$. («Kronecker Theorem».)

d) Let P be the set of prime numbers and $G =\mathbf{U}^P$. For every real number $T>0$, let $\lambda_T$ be the measure $T^{-1}\lambda$ on $[0,T]$, where $\lambda$ is the Lebesgue measure on $\mathbf{R}$. Let $\nu_T$ be the image measure of $\lambda_T$ by the continuous mapping $t\mapsto (p^{it})_{p\in P}$ from $[0,T]$ into G. When $T\rightarrow +\infty$, the family $(\nu_T)$ converges vaguely to $\mu_G$.