Finite additivity in outer measure Let $\{E_i\}_{i=1}^n$ be finitely many disjoint sets of real numbers (not necessarily Lebesgue measurable) and $E$ be the union of all these sets. Is it always true that 
$$
m^\star (E)=\sum_{i=1}^N m^\star(E_n)
$$
where $m^\star$ denotes the Lebesgue outer measure? If not, please give a counterexample. The Vitali set is a counterexample in the countable case, but I am not sure whether it is false in finite case.

It is a theorem of Lusin that the restriction of Lebesgue outer measure $\mu^*$ to any non null set X cannot be finitely additive. See the first theorem here: http://www.math.wisc.edu/~akumar/Thesis.pdf
