= Solution
Define $L_\xi(f)=\lim_NA_N(f)$. Limits preserve <linearity> and <positivity>, and $|L_\xi(f)|\leq\|f\|_\infty$, $L_\xi(1)=1$. The <Riesz-Markov-Kakutani representation theorem> supplies a unique Radon <probability measure> $\mu^\xi$. By its definition,
$$
\int f\,d\mu_N^\xi\longrightarrow\int f\,d\mu^\xi\quad(f\in C(\mathbb T)),
$$
which is exactly weak-star convergence in $C(\mathbb T)'$.
If $\xi/(2\pi)=p/q$ in lowest terms, the orbit has exactly $q$ points. Averaging a periodic sequence gives
$$
\boxed{\mu^\xi=\frac1q\sum_{r=0}^{q-1}\delta_{e^{ir\xi}}.}
$$
This includes $q=1$. If $\xi/\pi$ is irrational, then $e^{ik\xi}\ne1$ for every nonzero integer $k$. The preceding calculation gives $L_\xi(z^k)=0$ for $k\ne0$ and $L_\xi(1)=1$, exactly the integrals against normalized <Haar measure> $m$. <Density> of <trigonometric polynomials> implies equality on all <continuous functions>, and uniqueness gives
$$
\boxed{\mu^\xi=m\quad\text{when }\xi/\pi\text{ is irrational}.}
$$
These are the periodic-orbit and irrational-rotation cases of <equidistribution>.
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