Let $(X,\mathcal F,μ,T)$ be a dynamical system (i.e. μ is a probability measure and Τ is μ-preserving) and $\mathcal S\subset\mathcal F$ be a family of sets such that for any $A \in \mathcal F$ and $ε>0$ there exists a $B \in \mathcal S$ with $μ(A\triangle B)\lt ε$.
Assume that $\lim_{N\to \infty}\frac{1}{N}\sum_{n=0}^N μ(Α\cap T^{-n}B)>0$ , for all $A,B\in S$, with positive measure. Can we conclude that the system is Ergodic?
The motivation behind this question is to understand the concept of ergodicity better, by finding "minimal" conditions for it to occur.
I know, for example, that if the above limit is not only positive but equal to $μ(Α)μ(Β)$ then the system is ergodic and vice versa!
So I am thinking that it is natural to either construct a counterexample (which I tried to do, but failed) or to prove that if the limit is positive it necessarily equals $μ(Α)μ(Β)$.
Another fact; Since Von Neumann's mean ergodic theorem implies that $$\lim_{N\to \infty}\frac{1}{N}\sum_{n=0}^N μ(Α\cap T^{-n}B)=\int_A\mathbf{E}[1_B|\mathcal{I}]dμ$$ where $\mathbf{E}[1_B|\mathcal{I}]$ is the conditional expectation of $1_B$ with respect to the σ-algebra $\mathcal{I}=\{D\in \mathcal F: T^{-1}D=D\}$, then for the implication to be true it would suffice to show that if for all $A,B\in \mathcal S$ with positive measure we have that $\int_A\mathbf{E}[1_B|\mathcal{I}]dμ>0$, then we specifically have the equality $\int_A\mathbf{E}[1_B|\mathcal{I}]dμ=μ(Α)μ(Β)$.