Let $\alpha$ be an outer measure on $\mathcal P(\mathbb R^d)$, with the property that for every $A,B\subseteq \mathbb R^d$ with $A\cap B=\emptyset \Rightarrow \alpha(A\cup B)=\alpha(A)+\alpha(B)$.
Is $\alpha$ a measure on $\mathcal P(\mathbb R^d)$? I think yes. By induction I can show that $A_1,\dots,A_n$ disjoint, then $\alpha(\bigcup_{i=1}^n A_i)=\sum_{i=1}^n \alpha(A_i)$ but this works only for a finite number of sets.