Find sets $E_1, E_2,\dots$ of finite outer measure s.t. $E_k \searrow E$ and $\lim |E_k|_e > |E|_e$ I am trying to show existence of sets $E_1, E_2,\ldots$ s.t. $E_k \searrow E$, $|E_k|_e$ (outer measures of $E_k$'s) are finite and $\lim |E_k|_e > |E|_e$ strictly.
I took a nonmeasurable subset of $[0,1)$ and tried to find a sequence of nonmeasurable sets decreasing to empty set. We know that all these sets have positive outer measure, but is it possible that their limit is positive? (or is there such a decreasing sequence?)

The unit interval can be partitioned into countably many sets $B_k$ of outer measure 1.  (In fact, much more is true; it can be partitioned into continuum many Bernstein sets, i.e., sets such that every uncountable closed set meets them all.)  Let $E_n=\bigcup_{k>n}B_k$.  These sets $E_n$ all have outer measure 1 and they converge to the empty set.
