A ratio ergodic theorem on Borel actions of Zd and Rd - Holt, Eric Norman, author; Rudolph, Daniel, advisor (Colorado State University. Libraries, 2009)
We prove a ratio ergodic theorem for free Borel actions of Zd and Rd on a standard Borel probability space. The proof employs an extension of the Besicovitch Covering Lemma, as well as a notion of coarse dimension that originates in an upcoming paper of Hochman. Due to possible singularity of the measure, we cannot use functional analytic arguments and therefore diffuse the measure onto the orbits of the action. This diffused measure is denoted μx, and our averages are of the form 1/μx(Bn) ∫ Bn f o T-v(x)dμx. A Følner condition on the orbits of the action is shown, which is the main tool used in the proof of the ergodic theorem. Also, an extension of a known example of divergence of a ratio average is presented for which the action is both conservative and free.
Keywords: Besicovitch Covering Lemma; Borel actions; coarse dimension; ergodic theorem; Hochman; mathematics