Gagliardo-Nirenberg-Sobolev Inequality w.r.t. general probability measure

Question

The Gagliardo-Nirenberg-Sobolev inequality states that the $L^{p*}$ norm of a function with compact support is bounded by the $L^p$ norm of its derivative. Typically $L^p$ is in the sense of the Lebesgue integral on $\mathbb{\mathbb{R}^d}$. Can we get similar results if we consider a general probability measure $\mu$ on $\mathbb{R}^n$ in which the integration is with respect to $\mu$ instead of the Lebesgue measure?