I'm trying to understand the key proof of the parametrization trick of this paper "Auto-Encoding Variational Bayes" http://arxiv.org/pdf/1312.6114v10.pdf. It's section 2.4 on page 4.
which states that: Let $z$ be a continuous random variable, and $z ∼ q_\phi(z|x)$ be some conditional distribution. It is then often possible to express the random variable z as a deterministic variable $z = g_\phi(\epsilon, x)$, where $\epsilon$ is an auxiliary variable with independent marginal $p(\epsilon)$, and $g_\phi(.)$ is some vector-valued function parameterized by $\phi$...
... A proof is as follows. Given the deterministic mapping $z = g_\phi(\epsilon, x)$ we know that $q_\phi(z|x)\prod_idz_i = p(\epsilon)\prod_id\epsilon_i$. Therefore, $\int q_\phi(z|x)f(z) dz = \int p(\epsilon) f(z) d\epsilon = \int p(\epsilon)f(g_\phi(\epsilon, x)) d\epsilon$.
I don't quite understand how $q_\phi(z|x)\prod_idz_i = p(\epsilon)\prod_id\epsilon_i$ comes about. Any help is appreciated.