I'm studying the Aldous-Hoover Theorem and I've seen in many references that its analogous in dimension zero is a result of abstract measure theory:
Let $\nu:E\rightarrow [0,1]$ be a Borel probability measure on a standard Borel space $E$. Then there is a measurable $f:[0,1]\rightarrow E$ such that, given a random variable $U\sim U[0,1]$, \begin{equation*}f(U)=^d X,\end{equation*}where $X$ is a random variable with values in $E$ and law $\nu$.
I am interested in seeing the proof of this result. Can you tell me where to find it?
Thank you