This is a problem from Leo Breiman- probability. Given a process $X_1,X_2,...,$ abbreviate $\mathcal{F}_n=\mathcal{F}(X_n,X_{n+1},...)$. Show that the tail $\sigma$-field $\tau$ on the process has the zero-one property iff for every $A \in \mathcal{F}(X_1,X_{2},...)$ $$\lim_n \sup_{B \in \mathcal{F}_n}|P(A \cap B)-P(A)P(B)|=0 $$
We can use the fact that if $Z,X_1,...$ are random variables and $\tau$ is the tail $\sigma$-field on the process $X_1,X_2,...$. Then $$\mathbb{E}(Z|X_n,X_{n+1},...) \rightarrow_{a.s \text{ or first mean} } \mathbb{E}(Z|\tau)$$.
I'm having trouble with the $\Rightarrow$ part. This is kind of a converse of the zero-one law, but I know that the converse is not true in general. Actually the term limsup confuses me. I tried with $Z=1_A$ but then I don't know how to connect this with the limsup of the expression
Let $Y_n=E(I_A|\mathcal F_n)$. Then $(Y_n)$ is a uniformly integrable martingale and it converges a.s. and in $L^{1}$ to $I_A$. Now $|E[I_B E(I_A|\mathcal F_n)-P(A\cap B)]|=|E[I_B E(I_A|\mathcal F_n)-EI_B I_A| \leq E|Y_n-I_A|$ for all $B \in \mathcal F_n$. Take sup ober all $B \in \mathcal F_n$ to finish the proof.