Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states that for every $t \in [0,1]$, there exists a set $S \in \Sigma$ with $\mu(S) = t$.

I would like to use the following stronger conclusion for such a measure: 

> There exists a chain of sets $\{S_t \mid t \in [0,1]\}$ in $\Sigma$,
> with $S_t \subseteq S_r$ whenever $0 \leq s \leq r \leq 1$, such that
> $\mu(S_t) = t$ for all $t \in [0,1]$.

(One can view this as the existence a right inverse to the map $\mu \colon \Sigma \to [0,1]$ in the category of partially ordered sets.)

This statement appears (albeit hidden within a proof) on the Wikipedia page for "[Atom (measure theory)][1]," and even includes a sketch for the proof!  However, I would like to see some mention of this in the literature.  Can anyone provide me with such a reference?


  [1]: http://en.wikipedia.org/wiki/Atom_%28measure_theory%29#Non-atomic_measures