For amenable groups it should be known that ergodic actions cannot be strongly ergodic, so in particular we are not talking of actions of amenable groups. You are perfectly right.

Sorry, there was a typo in the question. Clearly you do not know a priori any convergence of the sequence $\sigma_k$ to a measurable map $\sigma$, because otherwise, as you pointed out, you could apply the ergodic property to the limit function and get that the limit is essentially constant.

A group $\Gamma$ acts strongly ergodically on a measure space $(X,\mu)$ if given a sequence of measurable sets $\{A_n\}_{n \in \mathbb{N}}$ such that $\lim_{n \to \infty} \mu(A_n \Delta \gamma.A_n)=0$, then it holds $\lim_{n \to \mathbb{N}} \mu(A_n)(1-\mu(A_n))=0$. Intuitively, any sequence of measurable sets which is asymptotically invariant must have either full or null measure at infinity.

Since the statement in the notes is not true, do you think there is an "easy" generalization of this construction valid for $\mathbb{C}^2$ to the more general case of $\mathbb{C}^n$?