Is it possible to define an order-preserving map $\mu: \mathcal{P}\left(\mathbb{R}^d\right)\to [0,+\infty]$ such that $\mu(\emptyset)=0$ and that there exist countable mutually disjoint non-empty sets $\{E_n: n\in\mathbb{N}\}$ with $ E_i \cap E_j=\emptyset$ for $i\neq j$, satisfying $\mu\left(\cup_n E_n\right)>\sum_n \mu(E_n)$, where the infinite sum converges?
Since countable subadditivity is included as part of the definition of an outer/exterior measure, I'm just curious if the opposite could be true. I've never seen one such counterexample anywhere. Hence the question.
Consider $\mu(A)=\begin{cases} 0 \text{ if }0,1\notin A,\\ 1 \text{ if }|A\cap\{0,1\}|=1,\\ 3 \text{ if } 0,1\in A. \end{cases}$