Bounded Pluriharmonic Functions and Holomorphic Functions on Teichmüller Space.

The article focuses on the boundary behavior of bounded pluriharmonic and bounded holomorphic functions on the Teichmüller space \( T_{g,m} \) of Riemann surfaces of finite analytic type. It establishes a version of Fatou's theorem, proving that every bounded pluriharmonic function admits radial limits almost everywhere along Teichmüller geodesic rays with respect to the pluriharmonic measure on the Bers boundary. Additionally, it presents a version of the F. and M. Riesz theorem, showing that the radial limit of a non-constant bounded holomorphic function cannot be constant on any measurable subset of positive pluriharmonic measure. As a corollary, the paper demonstrates the non-ergodicity of the Torelli group action on the space of projective measured foliations for closed surfaces of genus \( g \geq 2 \). These results connect complex function theory, hyperbolic geometry, and the topology of moduli spaces, contributing to the understanding of the interplay between analytic and geometric structures on Teichmüller space.