Characterizations of Measurability-Preserving Ergodic Transformations

Let (S, A, μ) be a finite measure space and let φ: S → S be a transformation which preserves the measure μ. The purpose of this paper is to give some (measure theoretical) necessary and sufficient conditions for the transformation φ to be measurability-preserving ergodic with respect to μ. The obtained results extend well-known results for invertible ergodic transformations and complement the previous work of R.E. Rice on measurability-preserving strong-mixing transformations. 2000 Mathematics Subject Classification. Primary: 28D0