We prove a version of a Theorem of Furstenberg in the setting of Mapping class groups. Thurston measure defines a smooth measure class on space of projectivized measured laminations For every measure $\nu$ in this measure class, we produce a measure $\mu$ with finite first moment on the mapping class group such that $\nu$ is the unique $\mu$-stationary measure. In particular, this gives an coding-free proof of the already known result that the Lyapunov spectrum of Kontsevich-Zorich cocycle on the principal stratum of quadratic differentials is simple. This is a joint work with Alex Eskin and Maryam Mirzakhani.