A nonmeasurable partition of the reals
It is shown that there are Vitali type Lebesgue nonmeasurable subsets V of, say, the real unit interval with outer measure of V being equal to any preassigned positive real ≤ 1 and with inner measure of V being always equal to 0.
Kemp, Paula. "A nonmeasurable partition of the reals." Bulletin of the Belgian Mathematical Society-Simon Stevin 8, no. 1 (2001): 83-86.
Bulletin of the Belgian Mathematical Society - Simon Stevin