Corollary 3

— Let $\rho$ and $\sigma$ be two alien complex measures; then $|\rho + \sigma| = |\rho| + |\sigma|$.

Denote by $v$ (resp. $w$) a universally measurable function of absolute value 1 such that $\rho = v \cdot |\rho|$ (resp. $\sigma = w \cdot |\sigma|$) (Cor. 3 of Th. 2), and by A a universally measurable set carrying $\rho$, such that $B = \mathbf{C}A$ carries $\sigma$ (Prop. 13); then also $\rho + \sigma = (v \varphi_A + w \varphi_B) \cdot (|\rho| + |\sigma|)$. Since the function $v \varphi_A + w \varphi_B$ has absolute value equal to 1, the corollary follows from Prop. 2.