Let $(X,\Sigma)$ be a measurable space. Let $\mu$ and $\nu$ be two measures on thereon. Are there any reasonable restrictions on these measures to ensure the existence of a (measurable) function, $f: X \to X$ such that $$ \nu = f_*\mu$$ where $f_*\mu$ is the push forward of $\mu$ with respect to $f$, i.e., $$\nu(A) = \mu(f^{-1}(A))$$ for all $A \in \Sigma$.

I understand this is a pretty unstructured question. Perhaps, if we require structure of $f$. For example, if $X = \mathbb{R}$ and we require $f$ to be monotone or linear?


If $X$ is any separable completely metrizable space, such as $\mathbb{R}$, a sufficient condition is that $\mu$ is atomless and $\mu(X)=\nu(X)<\infty$, see here for the argument. It should be quite clear that handling atoms of $\mu$ is not easy, the will lead to atoms in $\nu$.

Alternatively, it is sufficient that both $\mu$ and $\nu$ are infinite but $\sigma$-finite and $\mu$ is atomless. This case follows from the previous one.


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