Automorphism on the unit interval compatible with a measure preserving set function

Question

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 $y \in I$?