Proposition 10

Let $f, g$ be two positive universally measurable functions on $X$, let $a$ be a number $\geq 0$, and let $\mu$ and $\nu$ be two positive measures on $T$. Then:

a) $\Lambda(f + g) = \Lambda f + \Lambda g$, $\Lambda(af) = a\Lambda f$. b) If $\mu$ and $\nu$ belong to the domain of $\Lambda$, then so do $\mu + \nu$ and $a\mu$, and one has $(\mu + \nu)\Lambda = \mu\Lambda + \nu\Lambda$, $(a\mu)\Lambda = a(\mu\Lambda)$.

The only non-obvious point is that $\mu + \nu$ belongs to the domain of $\Lambda$, which is treated by observing that every positive measure bounded above by $\mu + \nu$ is of the form $\mu' + \nu'$, where $\mu' \leq \mu$, $\nu' \leq \nu$ (the 'decomposition lemma', Ch. II, §1, No. 1). See also the next proposition.