Let T be a positive L1-L8 contraction. We prove that the following statements are equivalent in constructive mathematics. (1) The projection in L2 on the space of invariant functions exists: (2) The sequence (Tn)n¿N Cesáro-converges in the L2 norm: (3) The sequence (Tn)n¿N Cesáro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem. As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop.