Let $(\Omega,f,p)$ be an abstract probability space. Consider the variables $$ X_n = 1_{A_n} - 1_{A_n^C} \in L^p(\Omega)$$ I am interested in charactarizing when $X_n \to 0$ weakly, meaning that $E[Y X_n]\to 0$ for any $Y \in L^q(\Omega).$
If we have that $P(A_n) = \frac{1}{2}$ and they are independent, then $(X_n)_n$ will become an orthonormal sequence. That is $E[X_n X_m] = 0 $ for $n \not = m$ and $E[X_n X_n] =1.$ By weakly compactness of the unit ball, $(X_n)_n$ must at least have a weakly convergent subsequence etc. I believe the weak limit of $X_n$ will be $0$ in this case. We also do not worry too much about what happens in the beginning, so I think it might suffice if one 'eventually approximately' has $P(A_n) = \frac{1}{2}$ and $A_n s$ are 'eventually' independent.
So far I have derived some necessary conditions$\colon$
For $Y=1,$ $E[Y X_n] = E[1_{A_n} - 1_{A_n^C}] = P(A_n) - P(A_n^C) \to 0.$ Hence one should have $P(A_n) \to \frac{1}{2}.$
For $Y=1_B,$ $$E[Y X_n] = E[1_B (1_{A_n} - 1_{A_n^C})] = E[1_{B\cap A_n} - 1_{B \cap A_n^C}] = P(B\cap A_n) - P(B \cap A_n^C) \to 0.$$ Hence one should have $P(B\cap A_n) \to \frac{P(B)}{2}.$ Another way of expressing this is that $E[1_B 1_{A_n}] \to \frac{E[1_B]}{2}.$ In other words, $E[1_B | A_n] \to E[1_B],$ which can be generalized as $E[Y| A_n] \to E[Y],$ for any $Y \in L^q(\Omega).$
Back to $P(B\cap A_n) \to \frac{P(B)}{2} \colon$ Intuitively, $A_n$ eventually divide all the sets equally. Another way of seeing this is that $P(B\cap A_n) - P(B) P(A_n) \to 0.$ If we consider this for $B= A_1,$ one has $P(A_1 \cap A_n) - P(A_1) P(A_n) \to 0.$ Similarly for $k_1 \leq \cdots \leq k_m,$ one would have $$ P(A_{k_1} \cap \cdots \cap A_{k_m}) - P(A_{k_1}) \cdots P(A_{k_m}) \to 0 $$ as $k_1 \to \infty.$ This looks like the notion of being eventually independent, or eventually approximately independent if there is one. I have searched for generalizations of Borel Cantelli lemma, yet have not encountered conditions replacing independence, which covers the above case. How can I obtain a complete chracterization of $X_n \to 0$ weakly (when $\Omega$ is nonatomic)?
Btw this reminds me of the coin-tossing experiment, where one tosses a fair coin indefinitely and $A_n$ is the event that the $n$th tossing is heads, or variations of this.