Ergodicity and mixing for a selected probability map

Let $(X,\mathcal{A},\mu ,T)$ be a probability-preserving system: $\mu (X)=1$, $T:X\to X$ is measurable, and $\mu (T^{-1}E)=\mu (E)$ for every $E\in \mathcal{A}$ (Measure-preserving transformations and systems). The ergodicity convention is that every strictly invariant set $E\in \mathcal{A}$, meaning $T^{-1}E=E$, has measure zero or one, as in Ergodicity relative to an invariant measure.

For $E,F\in \mathcal{A}$ and integer $n\ge 0$, put $d_n(E,F)=\mu (E\cap T^{-n}F)-\mu (E)\mu (F)\mathrm{.}$ The map is strongly mixing if $d_n(E,F)\to 0$ for every measurable pair $(E,F)$. It is weakly mixing if $\frac{1}{N}{\sum }_{n=0}^{N-1}\mathrm{\mid }{d}_n(E,F)\mathrm{\mid }⟶0$ for every such pair as $N\to \mathrm{\infty }$. Thus the absolute value is inside the Cesàro average, exactly as in Strong and weak mixing on a probability space. These are properties of the selected discrete map $T$ and its selected invariant probability $\mu$; no flow-level property is implicit.