Measurability – Preserving Weakly Mixing Transformations

In this paper we investigate measure-theoretic properties of the class of all weakly mixing transformations on a finite measure space which preserve measurability. The main result in this paper is the following theorem: If φ is a weakly mixing transformation on a finite measure space ( S, A , μ ) with the property that φ ( A ) ⊆ A , then for every $A, B $ in A there is a subset $J(A,B)$ of the set of non-negative integers of density zero such that lim m → ∞ ,m ∉ J(A,B) μ (A ∩ φ ᵐ(B)) = (μ (A) / μ (S))lim n → ∞ μ \,(φⁿ(B)). Furthermore, we show that for most useful measure spaces we can strengthen the preceding statement to obtain a set of density zero that works for all pairs of measurable sets A and $ B.$ As corollaries we obtain a number of inclusion theorems. The results presented here extend the well-known classical results (for invertible weakly mixing transformations), results of R. E. Rice [17] (for strongly mixing), a result of C. Sempi [19] (for weakly mixing) and previous results of the author [8, 10] (for weakly mixing and ergodicity). 2000 Mathematics Subject Classification. Primary: 28D05, 37A25; Secondary: 37A05, 47A35