4
$\begingroup$

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.

1 Answers 1