Outer and inner vanishing measures and division in $H^\infty + C$

Measures on the unit circle are well studied from the view of Fourier analysis. In this paper, we investigate measures from the view of Poisson integrals and of divisibility of singular inner functions in H^\infty + C . Especially, we study singular measures which have outer and inner vanishing measures. It is given two decompositions of a singular positive measure. As applications, it is studied division theorems in H^\infty + C .