Measurability – Preserving Weakly Mixing Transformations - Fatkić, Huse (Sarajevo Journal of Mathematics, 2024)
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 _m → ∞ ,m otin J(A,B) μ (A φ ^m(B)) = (μ (A) / μ (S))_n → ∞ μ (φ^n(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