I am facing a problem in measure theory and there is a technical part that is giving me a hard time:
As it is defined, $\limsup(A_n)=\{x: x \;\text{in infinitely many sets of the sequence} (A_n)\}$ and it is easy to see that an equivalent expression is $\limsup(A_n)=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$. I need to find a sequence of sets $(B_n)$ such that $\limsup(B_n)=\{x: x\;\text{in infinitely many sets of} (A_n)\;\text{such that if}\; x\in A_m\cap A_n,\;\text{then}\;|m-n|>k\}$ but I'm stuck; any ideas?
I assume you mean you want all elements $x$ such that $x$ occurs in infinitely many $A_n$ and also for each $n,m$ if $x\in A_m\cap A_n$ then $|m-n|>k$.
Then consider taking $B_n$ to be all $x$ such that $x\in A_n$ and for $j<n$ if $x\in A_n\cap A_j$ then $|j-n|>k$. This gets you the result you want I believe.