I am trying to understand the interpretation of tail-events. The current answers existing on this matter helped me but did not convince me fully (in particular https://math.stackexchange.com/a/1414842/114720).
In fact, the answer cited above says that any event belonging in the tail field is independent of any finite number of events.
Contrary to the answer cited above, I would rather say that an event belong to the tail-field of $A_1$, $A_2$... is independent on any finite set of events$\in \{A_k\}_{k=1,.. ,n}$ conditional of the knowledge of $A_n$, $A_{n+1}$.. for any n.
The reason for this came from considering the algebra of a finite set $\Omega = \{1,2,3,4\}$. If we take $A_1 = 4$, $A_2 = \{1,2,3\}$, $X_1 = \sigma(\{4\})$ and $X_2 = \sigma(\{1,2,3\})$, $X_2$ contains by complementarity the event $A_1 = \{4\}$, so an element of $X_2$ is (trivially) dependent on $A_1$.
Am I right or I am missing something ?
Thank you!