# Reverse engineering distributions

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.]

• 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. – lonza leggiera Apr 18 at 3:09
• 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.) – Gorgo Apr 18 at 14:28
• 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 $\mathbb Q(B) := \mathbb P(f(B))$. – Kolodez May 5 at 14:36
• 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 $Q(B) := P((f|_D)^{-1}(B))$. – Kolodez May 7 at 13:45