Let $(X, \mathfrak B, \mu)$ be a probability space and $T:X \to X$ be a measure preserving transformation.
Definition: $T$ is s.t.b ergodic if for any $E \in \mathfrak B$ with $T^{-1}E=E$ implies $\mu(E)=0$ or $1$.
Qn: Prove that $T$ is ergodic if for all $A,B \in \mathfrak B$; $\mu(A)\mu(B)>0$, $\exists n \geq 1$ (depending on A, B) such that $\mu(T^{-n}A \cap B)>0$.
Now if $E \in \mathfrak B$ is s.t $T^{-1}E=E$ and $\mu(E)>0$ then putting $A=E$ and $B=E^c$ we have $\mu(T^{-n}A \cap B)=0$ this implies that $\mu(E^c)=0$ So we are done right?