A constructive ergodic theorem

Abstract:

As discussed by Bishop, Birkhoff’s Ergodic Theorem is not constructively valid. In this paper we present an hypothesis which is necessary and sufficient for the constructive almost everywhere convergence of the Césaro averages of the translates of an integrable function by a measure preserving transformation. In addition necessary and sufficient conditions are given for the limit function to be constructively integrable. Also we present a necessary and sufficient condition that the averages converge to a constant function and give an equivalent formulation of this condition for finite measure spaces. Several interesting examples are given which satisfy these conditions.