If $\mu(X) = \infty$ and for all $F \in {\cal E}$, there is $E \in {\cal E}$ such that $0 < \mu(E) < \mu(F)$, then $\mu$ is onto.

Question

Suppose that $(X, {\cal E}, \mu)$ is an infinite measure space (meaning that $\mu(X) = \infty$) with the property that for every $F \in {\cal E}$ with $\mu(F) >0$, there is an $E \in {\cal E}$ such that $0 < \mu(E) < \mu(F)$. Then the range of $\mu$ is all of $[0,\infty]$.

Any ideas on where to get started?