# On the weak closure of convex sets of Probability measures

created by gigli on 17 Nov 2008
modified on 09 Feb 2010

[BibTeX]

Accepted Paper

Inserted: 17 nov 2008
Last Updated: 9 feb 2010

Journal: Rendiconti di matematica
Year: 2009

Abstract:

We prove that a $W_2$-closed, geodesically convex subset $\mathcal C$ of $P^r_2({R^d})$ is closed with respect to weak convergence in $P^r_2({R^d})$. This means that if $(\mu_n)\subset \mathcal C$ is such that $\mu_n\to\mu$ in duality with continuous bounded functions and $\sup_n \int x ^2d\mu_n<\infty$, then $\mu\in \mathcal C$ as well.

Keywords: Wasserstein distance, Geodesic convexity