Denote $\mu$ to be the Lebesgue measure on $(0,1]$, $\{K_{n}\}$ a sequence of sets, and $\mathcal{B}(0,1]$ the borel sigma algebra on $(0,1]$. What would be an example of an $\{K_{n}\} \subseteq \mathcal{B}(0,1]$ s.t. $\mu(\lim \inf) \neq \mu(\lim \sup)$?
I have been thinking about it for awhile now and can't come up with anything. The help would be appreciated!
Try with this: for $n\geq 2$ define $$A_n = \left\{ \begin{matrix} (0, \frac{1}{2}] & \mbox{ if $n$ is even} \\ [\frac{1}{3} - \frac{1}{n} , 1- \frac{1}{n}] & \mbox{ if $n$ is odd} \end{matrix} \right. $$
Then
$\liminf_n A_n = [\frac{1}{3} , \frac{1}{2}]$, so its measure is $\frac{1}{6}$
$\limsup_n A_n = (0,1)$, so its measure is $1$
But $\mu(A_n)$ oscillates between $\frac{1}{2}$ and $\frac{2}{3}$