Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. EDIT: to make the question easier, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$.
We define partial Fourier amplitudes
$$G_N(q)=\sum_{1\leq n\leq N} \epsilon_n \exp(inq)$$
and the corresponding intensities
$$S_N(q)=\frac{1}{N}|G_N(q)|^2$$
Does the limit
$$d\mu(q)=\lim_{N\to\infty} \left[ S_N(q) \frac{dq}{2\pi}\right]$$
make sense? How do we prove rigorously that it exists?
For context, this definition of $d\mu(q)$ (the Fourier intenstiy measure) comes from (2.3) in the following paper:
Luck, J. M. "Cantor spectra and scaling of gap widths in deterministic aperiodic systems." Physical Review B 39, no. 9 (1989): 5834.
This paper can be found here (paywalled)