Proposition 3

— For every family $(f_\iota)_{\iota \in I}$ of functions in $J_+$,

For every finite subset $J$ of $I$, it follows from Th. 2 (by induction on the number of elements of $J$) that $\mu^*\left( \sum_{\iota \in J} f_\iota \right) = \sum_{\iota \in J} \mu^*(f_\iota)$; as $J$ runs over the set of finite subsets of $I$, the functions $g_J = \sum_{\iota \in J} f_\iota$ belong to $J_+$ and form a directed set for the relation $\leq$, whose upper envelope is the function $\sum_{\iota \in I} f_\iota$; the proposition therefore follows from Th. 1.