We know that Lebesgue Measure possesses the following continuity property:
If ${\{B_k\}_{k=1}^{\infty}}$ is a descending collection of measurable sets and $m(B_1)<\infty$ , then \begin{equation} m\left( \bigcap_{k=1}^{\infty}B_k \right)=\lim_{k\rightarrow \infty}m(B_k) \end{equation} Now we allow the sets fail to be measurable. We consider the following problem.
Give an example of a sequence of descending sets $\{E_i\}_{i=1}^{\infty}$ and $m^*(E_1)< \infty$, such that \begin{equation} m^*\left(\bigcap_{i=1}^{\infty}E_i\right)<\lim_{i\rightarrow \infty}m^*(E_i) \end{equation}
I think the Vitali sets will be useful, but I don't know how to construct the example.