Let T be a positive L1-L∞ 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.