 Solution

Let $\alpha ={\Phi }_{\#}{\mu }_0={\Phi }_{\#}{\mu }_1$. The corrected radial disintegration of an orthogonally invariant measure, whose proof did not require absolute continuity of measures, gives for every test function $\eta$Finite Borel measures on Euclidean space are determined by their integrals against test functions: the difference defines the zero distribution, or one can approximate continuous compactly supported functions uniformly by smooth functions and use regularity of the measures. Therefore