Exercise on Lebesgue measure ( Treatise of Analysis Vol2 by Dieudonné) Someone challenge me to bring the solution from anywhere! So I have posted here and see, I am optimist because this website is excellent and its members are so helpful.
Let me start with this definition:

A measure $\mu$ on a space $X$ is called diffuse if $\mu({x})=0$.
Remark [Problem 7(d) & 13.9]: can be see in this part of the book


This is the exercise: P. 211
I have captured a picture scaring of misquoting the context:

 A: Hints (assuming you have a) already, and that $X$ is metrisable and hence Polish, as the hint for c) suggests):
For b), you want to have a map $f\colon X\to [0,1]$ such that $f^{-1}[[0,t)]={U_t}$, for all $t\in[0,1]$. What would $f^{-1}[\{t\}]$ be in this case? To show that $f$ is continuous, use the fact that $\bigcap_{t'>t}U_{t'}=\bigcap_{t'>t}\overline U_{t'}$ (so it's a closed set).
Notice that the continuous function in b) would be a homeomorphism if only it was injective (as both spaces are compact). We want to make a similar construction which will result in a homeomorphism at the expense of removing a null set.
To to this, we want to find a tree $T$ of open subsets of $X$ such that for any node $\sigma$ in the tree, there are finitely many sons which are disjoint open sets contained in $\sigma$ along with their closures, and which have collectively full measure in $\sigma$. Then you can recursively match each level of the tree to a finite family of disjoint open intervals in $[0,1]$, and the matching will induce the homeomorphism.
