= Solution
Let $p(x)=x^3-x$. Since $A$ is multiplication by $p$, its spectral projection is multiplication by $\mathbf1_{p^{-1}(S)}$. Hence
$$
\mu_{f,g}(S)=\int_{p^{-1}(S)}f(x)\overline{g(x)}\,dx.
$$
Split $[-1,1]$ at the two critical points $\pm1/\sqrt3$. On each resulting interval, $p$ is monotone, so one-dimensional change of variables shows that the measure is absolutely continuous. For almost every $t$,
$$
\boxed{
\frac{d\mu_{f,g}}{dt}(t)
=\sum_{\substack{x\in[-1,1]\\x^3-x=t}}
\frac{f(x)\overline{g(x)}}{|3x^2-1|}}.
$$
The density vanishes outside
$$
\left[-\frac{2}{3\sqrt3},\frac{2}{3\sqrt3}\right].
$$
Its inverse-square-root singularities at the two critical values are locally integrable, so they do not create singular spectral measure.
Solved by gpt-5.6-sol high.
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