Let $(X, T):= (\{0,1\}^{\mathbb Z}, \text{the shift transformation})$ be the full two shift system. Then the set $M(X, T)$ consisting of all invariant (probability) measures on that system is a compact metrizable space under the weak\* topology so that we can talk about residual subsets and meager subsets of $M(X, T)$.


Now we consider just three kinds of mixing properties: ergodicity, weak-mixing, strong-mixing. My question is whether the set of weak-mixing measures form a residual subset of $M(X, T)$ or a meager subset or otherwise and what is a reference containing the result.


For the other two properties, we know:


* The ergodic measures form a residual subset. ((21.9) in [DGS])
* The strong-mixing measures form a meager subset. ((21.13) in [DGS])


[DGS] Ergodic Theory on Compact Spaces - Denker & Grillenberger & Sigmund




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**Edit: Remarks regarding the answer**


The title of the paper referred in the selected answer is **"A note on mixing processes"**. The paper contains a proof of the fact that for the full shift ${\mathbb R}^{\mathbb Z}$ (with the alphabet being the reals), the weakly mixing measures on it form a dense $G\_\delta$ set (and hence a residual set). The proof works for the full shift with finite alphabet as well. If I understood the paper right, the weakly mixing measures on a (self-)homeomorphism of a compact metric space always form a G delta set, regardless of whether they are dense or not.



topological size of the set of weakly mixing measures on the full two-shift
Weak mixing is generic. The result is due to K. R. Parthasarathy, *"Indian Journal of Statistics"*, November 1962, Series A vol.24.


Note that in the measurable setting, this is due to Halmos (see his 1956 book *"lectures on ergodic theory"*).


