Proposition 9

— Let $(A_n)$ be a sequence of pairwise disjoint integrable sets such that $\sum_{n=1}^\infty |\mu|(A_n) < +\infty$; then

For, if $A = \bigcup_n A_n$ then $\varphi_A = \sum_{n=1}^\infty \varphi_{A_n}$, and the proposition follows from (17) and Cor. 2 of Th. 2 of No. 3.

The relation (18) is also expressed by saying that the measure $\mu$ is completely additive in the set of integrable subsets of $X$.