Let $\mu$ and $\nu$ be two Radon measures on a locally compact group $G$. For every irreducible unitary representation $\pi$ of $G$ and vectors $u$ and $v$ from the corresponding Hilbert space $H_\pi$ denote by $\pi_{u,v}$ the respective matrix coefficient, i.e. the function from $G$ to $\mathbb{C}$ that maps $g$ to $\langle u,\pi(g)v\rangle$.

Assume that $\int_G\pi_{u,v}(g)\mu(dg)=\int_G\pi_{u,v}(g)\nu(dg)$, i.e. the Fourier transforms of the meausures coincide. Can we conclude that $\mu=\nu$?

My idea to proove this is the following: The matrix coefficients of irreducible representations are closed under multiplication (since the tensor product of two irreducible representations is again irreducible) and complex conjugates (consider adjoint operators). Also the trivial representation always exists and by the Gelfand-Raikov theorem, the matrix coefficients separate points. Thus by a (more or less standard) application of the Stone-Weierstraß theorem, we can conclude that the measures coincide on compact sets and hence on all Borel sets (because of inner regularity).

My questions are:

(i) Is the reasoning I have sketched correct?

(ii) Does this result have a name?

(iii) Where do I find an explicit formulation of this result in the standard literature? E.g. it seems not to be mentioned in Folland's book on abstract harmonic analysis.

I am thankful for any kind of advice.