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)](http://en.wikipedia.org/wiki/Atom_%28measure_theory%29#Non-atomic_measures)," 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?




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**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.)



Reference for a strong intermediate value theorem for measures
I would say this is folklore (I proved it and used it many years ago on my undergrad thesis), but here is a concrete reference:


Such a family of measurable sets is called a $[0,1]$-family in *On the Skorokhod representation theorem* by Jean Carlos Cortissoz, PAMS, Vol.135, No. 12, 2007 (see Definition 4.1). A proof that such a family exists in any non-atomic space is given in Lemma 4.1. 


