Frédéric Le Roux has written a very lucid exposition of the Alpern genericity theorem. This theorem states that the same ergodic properties are generic in the set of volume-preserving measurable transformations and in the set of volume-preserving homeomorphisms.
More precisely, endow with the weak topology, i.e., the coarsest generated by , for all measurable subsets and equip with the uniform distance.
The theorem considers any invariant under volume-preserving measurable isomorphisms. It states that is a dense subset of if and only if is itself a dense subset of .