Proposition 6

The union of a finite family $(A_i)_{1 \leq i \leq n}$ of integrable sets is integrable, and

If, moreover, the $A_i$ are pairwise disjoint, then

For, if $A = \bigcup_{i=1}^n A_i$ then $\varphi_A = \sup \varphi_{A_i}$, therefore (§ 3, No. 5, Cor. of Prop. 12) if the $A_i$ are integrable then so is $A$; the relation (13) is a special case of the analogous relation for outer measures (§ 1, No. 4, Prop. 18), on taking account of the relation $|\mu|(A) = |\mu|^*(A)$; finally, if the $A_i$ are pairwise disjoint, then $\varphi_A = \sum_{i=1}^n \varphi_{A_i}$, whence (14).