As you know, kolmogorov $0-1$ law holds for a sequence of independent events, however, if the sequence is not independent, we may have events in the tail $\sigma$-algebra with non-trivial probability $p$ (that is, $0<p<1$).
I am trying to find such a counter-example where the sequence $\{A_n\}_{n=1}^\infty$ is a sequence of pairwise-independent events and the tail $\sigma$-algebra contains events with non-trivial probability.
Hints will be appreciated
Best Answer
let $X_j$ be independent Rademachers ($\pm 1$ with probability $1/2$ each). Consider the sequence $X_2,X_1X_2,X_3,X_1X_3,X_4,X_1X_4,...$. Then $X_1$ is measurable with respect to the tail σ-algebra while the random variables are pairwise independent. To upgrade this example to "events" from "random variables", just add $1$ to everything. – fedja