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.
As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop.
"A constructive view on ergodic theorems." J. Symbolic Logic 71 (2) 611 - 623, June 2006. https://doi.org/10.2178/jsl/1146620162