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Given a probability density function $f \colon \mathbb{R}^m \to \mathbb{R}^+$, and a measurable function $g \colon \mathbb{R}^m \to \mathbb{R}^n$, with $n \leq m$, I would like to know precise conditions on $g$ that imply that the (pushforward) probability measure on $\mathbb{R}^n$, induced by $g$ from $\mu_f$ (the probability measure determined by $f$), itself has a probability density function (i.e., is absolutely continuous relative to Lebesgue measure on $\mathbb{R}^n$). This should hold when, for example, $g$ has sufficiently good differentiability properties, due to the existence of an explicit formula for the density function as an integral, over an $(m-n)$-dimensional surface, of a formula involving partial derivatives of $g$ (the $n=m$ and $n=1$ versions of this formula can be found at the bottom of the Wikipedia page "Probability density function"). Is anyone able to tell me a good reference to the mathematics behind such formulas? Ideally, I would like to know where I can find such formulas stated as theorems with precise conditions on their applicability and with proofs of correctness.

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I have found an answer to my own question. In "Submersions and preimages of sets of measure zero" (Siberian Mathematical Journal 28:153-163), S.P. Ponomarev proves as Theorem 1 that if $\Omega \subseteq \mathbb{R}^m$ is open and $g \in C^r(\Omega,\mathbb{R}^n)$ where $r \geq m - n + 1$ then $g^{-1}$ preserves Lebesgue measure $0$ sets if and only if $g$ is almost everywhere a submersion (i.e., the Jacobian derivative $g'$ has rank $n$ almost everywhere). Thus, in the context of my question, a sufficient condition for the pushforward measure (induced by $g$ from $\mu_f$) to have probability density is the conjunction: $g \in C^r(\Omega,\mathbb{R}^n)$ on some measure-$1$ open subset $\Omega$, where $r \geq m - n + 1$, and $g$ is almost everywhere a submersion.

[In the Math review, MR0886871, of the paper cited, the parenthesized " $\Omega \subset \mathbb{R}^k$ " should read " open $\Omega \subset \mathbb{R}^n$ ".]

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