Corollary

*Let $f$ be a function $\geq 0$, $(g_n)$ a sequence of measurable functions $\geq 0$; then $\int^\bullet f \left( \sum_n g_n \right) d\mu = \sum_n \left( \int^\bullet f g_n\, d\mu \right)$.

For the case of a finite sequence, this is an immediate consequence of Prop. 2. The case of an infinite sequence may be deduced from this by means of Prop. 1, e).