Corollary

If $m$ and $m'$ are two majorizable measures on $T$, with values in $G$, then $I_{\Phi,m+m'} = I_{\Phi,m} + I_{\Phi,m'}$ and $I_{\Phi,\lambda m} = \lambda I_{\Phi,m}$ for every scalar $\lambda$.

The second assertion is immediate. The first signifies that for every function $f$ that is both $|m|$-integrable and $|m'|$-integrable,

The function $f$ is $(|m| + |m'|)$-integrable (Ch. V, §2, No. 2, Cor. 1 of Prop. 3), hence a fortiori $(|m + m'|)$-integrable, and the first member of (6) is indeed meaningful. To show (6), it suffices to do so for $f$ a step function over the $(|m| + |m'|)$-integrable sets, since the set of these functions is dense in $\mathscr{L}_F^1(|m| + |m'|)$ and the two members of (6) are continuous in the latter space, by virtue of (4). But for $f = a \varphi_X$, where $X$ is $(|m| + |m'|)$-integrable, the two members of (6) reduce to $\Phi(a, \int \varphi_X\,dm) + \Phi(a, \int \varphi_X\,dm')$, whence the corollary.