  We prove that the Birkhoff pointwise ergodic theorem and the Oseledets
multiplicative ergodic theorem hold for every flat surface in almost every
direction. The proofs rely on the strong law of large numbers, and on recent
rigidity results for the action of the upper triangular subgroup of SL(2,R) on
the moduli space of flat surfaces. Most of the results also use a theorem about
continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S.
Filip.
