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.]