If $E_1 \cup E_2$ is a disjoint union of subsets of $\mathbb{R}^d$, then $m^*(E_1 \cup E_2) = m^*(E_1) +m^*(E_2)$ may not hold

Question

On Stein's real analysis page 16, he says we cannot conclude that if $E_1 \cup E_2$ is a disjoint union of subsets of $\mathbb{R}^d$, then $m^*(E_1 \cup E_2) = m^*(E_1) +m^*(E_2)$. Any counterexample?

Answer 1

See exercise 33 in the book: Outer Measure of the complement of a Vitali Set in [0,1] equal to 1

Let $E_1$ be a Vitali set, $E_2 = [0,1] \setminus E_1$. Then $m^*(E_1) > 0$ and $m^*(E_2) = 1$, so $$m^*(E_1 \cup E_2) = 1 < m^*(E_1) + m^*(E_2).$$