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 ?


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