# Transformation between two measures

If $$\mu$$ and $$\nu$$ are two measures, both absolutely continuous with respect to the Lebesgue measure on $$\mathbb{R}^d$$ with smooth densities $$p_\mu(\mathbb{x})$$ and $$p_\nu(\mathbf{x})$$, does it always exist a smooth map $$f:\mathbb{R}^d \to \mathbb{R}^d$$ that transforms $$\mu$$ to $$\nu$$, i.e. $$\nu$$ is the pushforward measure of $$\mu$$ for some smooth map $$f$$ ? Can this $$f$$ be chosen to be invertible ?