By measure I mean a set function $\mu:\mathscr{P}(X)\rightarrow[0,\infty]$ which is
$\mu(\emptyset)=0$.
Subadditive: $\mu(A\cup B)\leq\mu(A)+\mu(B)$.
Fatou's lemma: $\mu(\liminf_{n}E_{n})\leq\liminf_{n}\mu(E_{n})$.
Note that the Fatou's lemma entails the countably subadditivity as well: \begin{align*} \mu\left(\bigcup_{n}E_{n}\right)\leq\sum_{n}\mu(E_{n}). \end{align*}
Also note that I do not assume the additivity that $\mu(A\cup B)=\mu(A)+\mu(B)$ for disjoint sets $A,B$.
Assuming that $\mu$ is nonatomic in the sense that if $\mu(E)>0$ then there is an $F\subseteq E$ such that $\mu(E)>\mu(F)>0$, then can we deduce that for every $0<t<\mu(E)$, there is an $F\subseteq E$ such that $\mu(F)=t$?
If you look at this post, the very crucial step uses additivity property in completing the proof. So I wonder if there is any counterexample for my guess.
