I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want some probability measures on $S^1$ which is obtained through a kernel, like Poisson kernel for the harmonic measures etc. Are there any theorems that the probabilty measures are generated by some kernel if they are 'good', i.e. they have no atoms, they may be absolutely continuous with respect to Lebesgue measure.

Please suggest me some references in this topic.

Also, are there good results regarding homeomorphisms of $S^1, S^n$ which are more explicit in nature/expression ?