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Suppose I am given a measurable function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and a probability distribution $\mathbb{P}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$. Assume that the support of $\mathbb{P}$ is a subset of the image of $f$. Is there always a distribution on $\mathbb{Q}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$ such that for a random variable $\mathbf{X}\sim \mathbb{Q}$, $$f(\mathbf{X}) \sim \mathbb{P}?$$

[I'm asking this question with a specific $f$ and $\mathbb{P}$ in mind. In this example, $\mathbb{P}$ has a density with respect to the Borel and Lebesgue measure. If that additional assumption is useful, feel free to impose it. But I think this problem is interesting more generally and I hope for a general theorem. To clarify, I am not asking about uniqueness or computation of the distribution of $\mathbb{Q}$. In my example, this would be a nightmare even if it was possible.]

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  • $\begingroup$ Is the $\ \mathbb{n}\ $ in $f:\mathbb{R^n}\rightarrow \mathbb{R}^n\ $ supposed to be $\ n\ $? If not, $\ \mathbb{Q}\ $ would have to be a distribution on a $\sigma$-algebra of $\ \mathbb{R^n}\ $ for the composition of $\ f\ $ with $\ \mathbf{X}\ $ to make sense. $\endgroup$ – lonza leggiera Apr 18 at 3:09
  • $\begingroup$ Hi Ionza, wow, that's a good spot! Yes, they are meant to be the same. (I wrote \mathbb{R^n} instead of \mathbb{R}^n once. Just edited the question.) $\endgroup$ – Gorgo Apr 18 at 14:28
  • $\begingroup$ Your question can be reduced in a straight-forward way to the question whether there is a prob. measure $\mathbb Q$ on $\mathcal B^n$ such that the forward measure $\mathbb Q_f$ is equal to $\mathbb P$, i.e. for all $A \in \mathcal B^n$, it is $\mathbb P(A) = \mathbb Q(f^{-1}(A))$. If your $\mathbb P$ has a density, you can plug it into this equation. Then you can try to look if you can use the structure of $f$ in any way. For instance, if $f$ is injective and its inverse function is measurable, you can simply define $\mathbb Q(B) := \mathbb P(f(B))$. $\endgroup$ – Kolodez May 5 at 14:36
  • $\begingroup$ If your function is not injective, but can you can find a $D \in \mathcal B^n $ such that $f|_D$ is injective, $f(D) = f(\mathbb R^n)$ and the inverse function of $f|_D$ is measurable, you can just define $Q(B) := P((f|_D)^{-1}(B))$. $\endgroup$ – Kolodez May 7 at 13:45

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