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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 $Auto(T^d)$ of volume-preserving measurable transformations and in the set $Homeo(T^d)$ of volume-preserving  homeomorphisms.
More precisely,  endow $Auto(T^d)$   with the weak topology, i.e., the coarsest generated by $\mu\mapsto\mu(A)$, for all measurable subsets $A$ and equip $Homeo(T^d)$  with the uniform distance.
The theorem considers any $P\subset Auto(T^d)$  invariant under volume-preserving measurable isomorphisms. It states that $P$  is a dense $G_\delta$ subset of $Auto(T^d)$ if and only if $P\cap Homeo(T^d)$ is itself a dense $G_\delta$ subset of $Homeo(T^d)$.