A generalization of a theorem of Atkinson to noninvariant measures

Daniel Ullman

Vol. 130 (1987), No. 1, 187–193

DOI: 10.2140/pjm.1987.130.187

Abstract

We prove that, if T is an ergodic, conservative, non-singular automorphism of a Lebesgue space (X,μ), then the following are equivalent for f in L¹(μ):

1. If μ(B) > 0 and 𝜖 > 0, then there is an integer n≠0 such that

2. liminf _n→∞∑ _j=0ⁿ⁻¹f(T^jx) ⋅(x) = 0 for a.e. x.
3. ∫ f dμ = 0.

Mathematical Subject Classification 2000

Primary: 28D05

Milestones

Received: 3 May 1986

Revised: 15 July 1986

Published: 1 November 1987