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.

I found this proof in Real Analysis by Royden:
Theorem: There are disjoint sets of real numbers A and B for which
m*(A U B) < m*(A) + m*(B).
Proof We prove this by contradiction. Assume m*(AU B) = m*(A) + m*(B) for every
disjoint pair of sets A and B. Then, by the very definition of measurable set, every set must be measurable. This contradicts the existence of a non-measurable set.
