Measures induced by analytic functions and a problem of Walter Rudin

The measure $\mu_\varphi$ induced by a bounded analytic function $\varphi$on the unit disk $U$ may be defined by $\mu_\varphi(E)=m(\varphi^{-1}(E))$, where $m$ is normalized Lebesgue measure on $\partial U$. We discuss the problem of characterizing such measures, and produce some necessary conditions which we conjecture are sufficient. Our main results are a construction showing that our conjectured sufficient conditions are sufficient for a measure to be weakly approximable by induced measures, and a construction of a function $\varphi$, not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies $\int\varphi\left(e^{i\theta}\right)^m\overline{\varphi \left(e^{i\theta}\right)^n} \frac{d\theta}{2\pi}=0$ if $m\ne n$, thus answering a question posed by Walter Rudin.