Let $\{E_n\}$ be a sequence of Borel sets on $[0,1]$. I want to construct an example of $m(\cap_{n=1}^\infty\cup_{k=n}^\infty E_k)>\lim\sup m(E_n)$.
I can prove in general $m(\cap_{n=1}^\infty\cup_{k=n}^\infty E_k)\ge\lim\sup m(E_n)$. For the example of $>$, I am trying to find a sequence of $E_n$ s.t. $\lim m(E_n)=0$ and $m(\cup_{k=n}^\infty E_k)=1$, but so far I can't construct such an example.
Best Answer
One way to do it is to find sets so that $\cup_{k=n}^\infty E_k = [0,1]$ for all $n$, yet $m(E_n) \to 0$. Try the sequence $$[0,1/2],\ [1/2,1],\ [0,1/3],\ [1/3,2/3],\ [2/3,1],\ [0,1/4],\ldots$$