Consider a sequence of $\sigma$-algebras $\mathcal{F}\_1,\mathcal{F}\_2,\dots$. Is it true that for any event $B$ in the tail $\sigma$-algebra $\mathcal{F\_{\text{Tail}}}$, it can be expressed as the $\limsup$ of some sequence of events $A\_1,A\_2,\dots$ such that $A\_i \in \mathcal{F}\_i$ for all $i$? Unfortunately, I have absolutely no intuition in this problem, and thus have no idea how to proceed.


Any help provided would be appreciated.



When are events in tail $\sigma$-algebra the limsup of some sequence of events?
No, let $\mathcal S\_0=\{\{0,1\},\emptyset\}$ and let
$\mathcal S\_1$ be the powerset of $\{0,1\}$. Let 
$\mathcal F\_n=\mathcal S\_0^{n-1}\times\mathcal S\_1
\times\mathcal S\_0^{\infty}$.


Let us write $X\in\{0,1\}^{\mathbb N}$ as $(X\_n)\_n$.


Let $B=\{X:X\_n=1$ for at most finitely many $n\}$. Suppose $B=$ lim sup $A\_n$. Then 



> 
> $X\_n=1$ for at most finitely many $n$
> $\iff$ infinitely many $A\_n$ occur.
> 
> 
> 


Case 1: for infinitely many $n$, $A\_n=\{0,1\}$.
Then $X\_n=1$ for all $n$ is a counterexample.


I'll let you think about the other cases.


