On some generic classes of ergodic measure preserving transformations

We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T T with property  A \mathbf {A} , a generic extension T ^ \widehat {T} of T T also has property  A \mathbf {A} . Here A \mathbf {A} stands for each of the following properties: (i) having the same entropy as T T , (ii) Bernoulli, (iii)  K, and (iv) loosely Bernoulli.