Let M denote the finite strictly positive measures on the cirlce \({\mathbb{T}}\). The author proves that for \(| z_ 0| <1\), f in the disc algebra and \(f(z_ 0)\not\in f(T)\) then for every \(\mu\) in M there is an interval \(I\subset T\) such that \[ f(z_ 0)=\frac{1}{\mu (I)}\int_{I}fd\mu. \] The proof involves a study of winding numbers. He gives examples to prove that some of the hypothesis can not be weakened.