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The pointwise ergodic theorem in subsystems of second-order arithmetic

Published online by Cambridge University Press:  12 March 2014

Ksenija Simic
Affiliation:
Department of Mathematics, University of Arizona, 617 North Santa Rita, Tucson, AZ 85721, USA. E-mail: ksimic@math.arizona.edu
Corresponding
E-mail address:

Abstract

The pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA0; moreover, the pointwise ergodic theorem is equivalent to (ACA) over the base theory RCA0.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

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