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Cross-posting from math stack-exchange since it's not getting any visibility there.

I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \subseteq F([0, z])$ for $y \le z$. Here $\lambda$ is the Lebesgue measure on the unit interval $I$ and $\Sigma(I)$ denotes the completion of the Borel $\sigma$-algebra on $I$. How do I show that there exists a measurable bijection $f: I \to I$ with $f([0, y]) = F([0, y])$ for all $y \in I$? What about just a measurable function?

One naive idea is to consider dyadic expansion of $I$. Each $x \in I$ lives in one nested infinite sequence of such dyadic subsets, say $I_j(x)$, $j=1, 2, \ldots$. We simply define $f(x) = \lim_j \sup \{y: y \in F(I_j(x))\}$. Note that the outer limit exists because the sequence is monotone non-increasing. This however may lead to pathological example with $f(x) \equiv 1$.

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Are you sure that you are not missing an hypothesis? If I take $$F([0,y]) = [0,y] \setminus {\bf Q}$$ there can be no such bijective $f$ because then $$f(\{y\}) = f([0,y]) \setminus \bigcup_{z<y} f([0,z]) = ([0,y]\setminus {\bf Q}) \setminus \bigcup ([0,z]\setminus {\bf Q}) = \{y\}$$ and thus $f$ would be the identity.

You may want to look at your objects modulo sets of zero measure. See e.g. Glasner, "ergodic theory via joinings" th. 2.13 for the conditions under which a function between two $\sigma$-algebras comes from a function between the underlying spaces. In short, you need $F(A\cup B) = F(A) \cup F(B) $ and $F(A^c) = F(A)^c$, together with measure preservation.

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  • $\begingroup$ Thank you very much for the reference! I knew the result could only be true under some equivalence relation but wasn't sure how to state it. I guess alternatively to the condition in Glasner's book, I could require $f$ to differ from $F$ only on a set of measure zero. Do you have other list of recommended readings in ergodic theory? $\endgroup$ – John Jiang Dec 10 '16 at 22:33
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    $\begingroup$ P. Walters, "an introduction to ergodic theory" and K. Petersen, "ergodic theory" are standard books that are well-written. $\endgroup$ – coudy Dec 12 '16 at 15:47

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