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)," and even includes a sketch for the proof! However, I would like to see some mention of this in the literature. I've checked the Wiki references and they both seem to prove the weaker statement. I looked in Fremiln's Measure Theory, vol. 2, and again found the weaker version but not the stronger.
Question: Can anyone provide me with such a reference?
A proof. In case anyone stumbles to this page and wants to see a proof, I'll sketch one that is more constructive than the one that I linked to above. Set $S_0 = \varnothing$ and $S_1 = X$. By Sierpiński, there exists $S_{1/2} \in \Sigma$ of measure $1/2$. For each Dyadic rational $q = m/2^n \in [0,1]$ ($1 \leq m \leq 2^n$), we may proceed by induction on $n$ to construct each $S_q$. Now given $r \in [0,1]$, set $S_r = \bigcup_{q \leq r} S_q$. (This is essentially the same method of proof as the one in the reference provided in Ramiro de la Vega's answer.)
