Example

Let us take for $X$ the compact space $\mathbf{U}$ of complex numbers of absolute value 1. On setting $\mu(f) = \int_0^1 f(e^{2i\pi t}) \, dt$ for every $f \in \mathscr{H}(\mathbf{U})$, one defines a positive measure of mass 1 on $\mathbf{U}$. On the other hand, let $\theta$ be a real number; for every integer $n \geq 0$, let $\nu_n$ be the unit mass placed at the point $e^{2i\pi n \theta}$ of $\mathbf{U}$, and let

so that $\mu_n$ is a positive measure of mass 1 on $\mathbf{U}$. Then, if $\theta$ is irrational, $\mu_n$ tends vaguely to $\mu$. For, since the linear combinations of the functions $z \mapsto z^k$ ($k \in \mathbf{Z}$) are dense in $\mathscr{H}(\mathbf{U})$ (GT, X, §4, No. 4, Prop. 8), it suffices to prove that $\mu_n(z^k)$ tends to $\mu(z^k)$ for $k \in \mathbf{Z}$. Now, for $k = 0$, $\mu_n(z^k) = \mu(z^k) = 1$; for $k \neq 0$, Since $e^{2i\pi k \theta} \neq 1$ (because $\theta$ is irrational), we infer that

therefore $\mu_n(z^k)$ tends to $0 = \mu(z^k)$. Under these conditions, Proposition 22 can be applied, and we see in particular that if $A$ is a subset of $U$ with negligible boundary with respect to $\mu$, then $\mu_n(A)$ tends to $\mu(A)$. In other words, if $p_n$ denotes the number of integers $k \in [0, n]$ such that $e^{2i\pi k \theta} \in A$, then $n^{-1} p_n$ tends to $\mu(A)$ as $n$ tends to $+\infty$.