Generating the terminal $\sigma$-algebra

Question

1. Given a (say stationary) sequence of random variables $X_1,X_2,\dots$. Denote by $T_n = \sigma(X_n,X_{n+1},\dots)$ and by $T= \bigcap_{n\in\mathbb N} T_n$, the terminal $\sigma$-algebra. If $A\in T$, is there a sequence of numbers $(i_k)\in\mathbb N^\mathbb N$ and of measurable bounded functions $(f_k) \in \prod_{k\in\mathbb N} L^\infty(\mathbb R^{i_k})$ such that $\textbf 1_A = \lim_{k\rightarrow \infty} f_k(X_{k},\dots,X_{k+i_k})$?
2. If yes, can we choose $i_k = i$ constant?
3. If no for 1., how could we approximate $1_A$ in terms of finitely many members of the sequence $(X_n)$? Do we know an interesting, explicit intersection-stable generator of $T$?