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Let $I = [0,1]$ be the unit interval. Let $\pi: I \to I$ be a Borel-measurable surjective map. Is the pushforward operator $\pi_*: \mathcal P(I) \to \mathcal P(I)$ surjective as well, where $\mathcal P(I)$ is the collection of Borel probability measures on $I$? In other words, given a probability measure $\nu \in P(I)$, does there exist a probability measure $\mu \in P(I)$ whose pushforward measure is $\nu$?

A related thread which deals with the continuous case.

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I have posted the question to MO:

https://mathoverflow.net/questions/196605/does-a-surjective-measurable-map-induce-a-surjective-pushforward-operator

and the answer there refers to a book in which the proof is done by relying on a uniformization result in the obvious way. To be a bit more specific, it uses the existence of a universally measurable right-inverse of $\pi$.

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