Let $G$ be a connected compact semisimple Lie group. Let $V$ be a faithful representation of $G$, with character $\chi \colon G \to \mathbb{C}$.
Let $\mu_G$ be the normalized left Haar measure. (So $\mu_G(G) = 1$.) We can consider the pushforward of $\mu_G$ along $\chi$. This gives a measure $\chi_* \mu_G$ on $\mathbb{C}$ whose support is the compact set $\chi(G)$.
Question. Does the measure $\chi_*\mu_G$ uniquely determine the character $\chi$?
Remarks. (i) Note that this is false if $G = S^1$. If $\chi_n$ is the character $z \mapsto z^n$ ($n \ne 0$), then $\chi_{n,*}\mu_{S^1}$ is the same as $\chi_{1,*}\mu_{S^1}$. So all non-trivial characters give rise to the same measure. In particular, it seems important to require that $G$ is semisimple.
(ii) I am not sure if the hypothesis "connected" is necessary. But my gut feeling says that things might go horribly wrong if $G$ is a non-trivial finite group.
(iii) Similarly, I don't know if the hypothesis "compact" is necessary. (Of course, if one removes this hypothesis, then it the claim that $\chi_*\mu_G$ has compact support is no longer true.)