Weakly mixing but not mixing quasi-Markovian processes

Let (f,α) be the process given by an endomorphism f and by a finite partition ${\displaystyle \alpha ={\left\{A_i\right\}}_{i=1}^s}$ of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if ${\displaystyle E(f,\alpha )\subset \{g:{\bigvee }_{{\left\{B_i\right\}}_{i=1}^s}\supset pg={\bigcup }_{i=1}^s{A}_i\times B_i\}}$. We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the 'chequer-wise' partition for σ × S, where σ is a Bernoulli shift and S is a weakly mixing automorphism, consists of constants.