I am wondering whether the following result is true:



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> Let $\mathcal W\_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W\_p(\mathbb R^d)$, such that supp($\eta$) $\subset$ supp($\gamma$). Then there is a sequence of probability measures $(\eta\_n)$ that converges to $\eta$ in $\mathcal W\_p(\mathbb R^d)$ and such that $\eta\_n$ is absolutely continuous w.r.t. $\gamma$, with $\frac{d\eta\_n}{d\gamma}$ being bounded.
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It seems to be a well-known result but I was not able to prove it nor to find a reference. Any help and suggestion are appreciated!



Density in Wasserstein space
First approximate $\eta$ with finite convex combinations of $\delta$-measures at points from $\mathrm{supp}\gamma$, then approximate each of these $\delta$-measures with measures with constant density with respect to $\gamma$ (just take the normalized restrictions of $\gamma$ to small balls). 


