In this talk, I will give the recent progress of my research on the function theory on Teichmüller space. We first start with the background of the research and recall the characterization of the pluriharmonic measure on the Bers slice (in the sense of Demailly) in terms of the Thurston measure. Then, I will discuss versions of the Fatou theorem and the F. and M. Riesz theorem for the Teichmüller space. As an application of these results, I will show that the action of the Torelli group on the space of projective measured foliations is not ergodic.

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