TITLE: Find sets $E_1, E_2,\dots$ of finite outer measure s.t. $E_k \searrow E$ and $\lim |E_k|_e > |E|_e$
QUESTION [7 upvotes]: 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?)

REPLY [4 votes]: The following paper shows that one can partition a closed set of positive measure in any number $\kappa$ of unmeasurable sets with full outer and inner measure, as long as $2\leq\kappa\leq\mathfrak{c}$. 
Alexander Abian, Partition of nondenumerable closed sets of reals, Czechoslovak Mathematical Journal, Vol. 26 (1976), No. 2, 207--210.
For the rest, proceed as in the answer of Andreas Blass.