Exercise 1, § 6

Let $f$ and $g$ be two measurable numerical functions, such that $a = M_\infty(f)$ and $b = M_\infty(g)$ are finite; show that, in order that $M_\infty(f + g) = M_\infty(f) + M_\infty(g)$, it is necessary and sufficient that, for every pair of real numbers $\alpha, \beta$ such that $\alpha < a ,\ \beta < b$, the set of $x \in X$ such that both $\alpha \leq f(x)$ and $\beta \leq g(x)$ not be locally negligible.