Give an example of a sequence $(E_n)_{n \in \Bbb N}$ and a measure $\mu$ so that $\mu(\liminf E_n) < \liminf \mu(E_n) < \limsup \mu(E_n) < \mu(\limsup E_n)$
This is a question on a past measure theory course I'm taking and I'm not sure how to answer. Could someone help me out? Thanks in advance.
Consider $\mu=\frac{1}{3}\delta_0 + \frac{1}{2}\delta_{\tfrac12}+\frac16\delta_{1}$ on $\Big([0,1],\mathscr{B}([0,1])\Big)$. Let $A_{2n+1}=[0,\tfrac12]$ and $A_{2n}=[\tfrac12,1]$, $n\in\mathbb{N}$.
$\liminf_nA_n=\{\tfrac12\}$, $\limsup_nA_n=[0,1]$.
$$\frac12=\mu\Big(\liminf_n A_n\Big)<\liminf_n\mu(A_n)=\frac23<\frac56=\limsup_n\mu(A_n)<1=\mu(\limsup_n(A_n)$$
One can also use Lebesgue measure on $[0,1]$ and have $A_{2n+1}=[0,1-b]$ and $A_{2n}=[a,1]$ with some choice of $a$ and $b$ to get the desired properties.