If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. However, I haven't been able to find a proof.
In his book, Halmos proves that the set of weakly-mixing transformations is a $G_\delta$ set and from there concludes that $\mathcal{E}$ is comeagre. Then he says "By a technique to the one used in the second category theorem it can also be proven that the set of all ergodic transformation is a dense $G_\delta$; this is, if anything, easier than the corresponding fact for weakly mixing transformations." However, in the proof he uses a characterization of weakly-mixing transformations and I don't know a generalization that works for ergodicity.
So my question is: Why is $\mathcal{E}$ a $G_\delta$ set in $G$? Does anyone know where I can find a proof of this theorem? Or do they know how to prove it? Thanks to anyone who answers in advance.