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My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and the book gradient flows in Metric Spaces by Amrosio-Gigli-Savare). In this setting one considers the space of probability densities $\rho\in \mathcal P_2(\mathbb R^d)$ (finite second moments) endowed with the quadratic Wasserstein metric $\mathcal W_2$. This metric space $(\mathcal P^2,\mathcal W_2)$ enjoys a nice differential structure. More precisely, geodesics $t\mapsto \rho_t\in \mathcal P_2$ can be nicely represented in terms of continuity equations $$ \partial_t \rho_t+\operatorname{div}(\rho_t v_t)=0, $$ where the vector field $v_t\in L^2(\mathbb R^d,d\rho_t)$ for a.e. $t$. I am somehow trying to identify the vector field $v_t$ in a particular context that I shall not describe here.

My specific question: assume I have a fixed probability measure $\rho \in \mathcal P_2$ which is absolutely continuous w.r.t Lebesgue $d\rho\ll dx$, i-e $d\rho(x)=\rho(x)dx$, and a fixed vector field $v\in L^2(\mathbb R^d,d\rho)$. Further assume that there is a second $d\rho$ measurable vector field $u(x)$ such that, for any smooth compactly supported vector field $\zeta\in \mathcal C^{\infty}_c(\mathbb R^d;\mathbb R^d)$, there holds $$ \int_{\mathbb R^d}\left<u(x),\zeta(x)\right>d\rho(x)=\int_{\mathbb R^d}\left<v(x),\zeta(x)\right>d\rho(x). $$ I would like to conclude that $u=v$ in $L^2(\mathbb R^d,d\rho)$, but for that I need to know that $C^{\infty}_c$ is dense in that space. What conditions do I need on $d\rho(x)=\rho(x)dx$ in order to ensure this density? I would be interested in optimal conditions, but any sufficient condition will do as well. Also, what if I relax $d\rho(x)\ll dx$, i-e if I don't assume that $d\rho$ is absolutely continuous?

Thank you in advance for your input and comments.

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    $\begingroup$ $C_0^{\infty}$ is dense in $L^2$ for any Borel measure $\rho$. $\endgroup$ Aug 12, 2014 at 15:58
  • $\begingroup$ OK, thanks @Christian Remling. Any specific reference I could cite? I was suspecting that, but for example I'm having a hard time figuring out what $L^2(\mathbb R^d,\delta_0)$ looks like (Dirac mass)... $\endgroup$ Aug 12, 2014 at 17:09
  • $\begingroup$ I know that Folland's Real Analysis text treats this, but it should be in any basic text really. $\endgroup$ Aug 12, 2014 at 17:38
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    $\begingroup$ @leomonsaingeon, I may be missing the point, but I think that $L^2(\mathbb R^d, \delta_0)$ is just $\mathbb C$, via the evaluation-at-0 map. $\endgroup$
    – LSpice
    Aug 24, 2014 at 13:14

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