We study completeness of the spaces $\mathcal{ P}_s^=$ of probability measures in $\mathbb{ R}^N$ which have equal (prescribed) moments up to order $s \in \mathbb{ N}$, endowed with the metric $d_s(\mu,\nu)=\sup_{ x \in \mathbb{ R}^N\setminus 0}\frac{ |\hat \mu(x)-\hat \nu(x)|}{ |x|^s}$, where $\hat \mu$ is the characteristic function of $\mu$. We prove that the spaces $(\mathcal{ P}_s^=,d_s)$ are complete if $s$ is even and construct suitable counterexamples to completeness for all odd $s$. This solves an open problem formulated by J. Carrillo and G. Toscani in 2007.
Categories: math.PR math.CA
