I am interested in a statement in the following math StackExchange thread https://mathoverflow.net/questions/127898/obtaining-conditional-probabilities-as-pushforwards-of-0-1
It is standard that every Borel probability measure on a polish space X can be obtained as pushforward of the uniform measure λ on [0,1] along an almost-everywhere-defined Borel-measurable function d:[0,1]→X . (In fact, d can always be taken to be continuous on a measure-1 $G_\delta$ subset...)
Do you know that is the "standard" result the author is referring to?
The case $X=\mathbb{R}$ you can find in many textbooks, e.g. section 2.5.2 of Resnick's Probability Path or Theorem 1.2.2 of Durrett's Probability: Theory and Examples. The idea is to let $F(x) = \mu((-\infty, x])$ be the cdf of the measure $\mu$ on $\mathbb{R}$, and then letting $d(y) = \inf\{s : F(s) \ge y\}$, which is more or less the "inverse" of $F$, will do the job. Note that $d$ is continuous except at countably many points.
For a general Polish space, use the fact that any Polish space $X$ admits a bimeasurable bijection $\phi : \mathbb{R} \to X$ (see Durrett Theorem 2.1.15 or any descriptive set theory book), and apply the previous case to the pullback $\mu'(A) = \mu(\phi(A))$.
I don't know off the top of my head how to prove, in the latter case, that $d$ can be taken a.e. continuous, but presumably you just have to be more careful in choosing $\phi$.