Lebesgue decomposition in action via semidefinite relaxations

Abstract

Given all (finite) moments of two measures $\mu$ and $\lambda$ on $\R^n$, we provide a numerical scheme to obtain the Lebesgue decomposition $\mu=\nu+\psi$ with $\nu\ll\lambda$ and $\psi\perp\lambda$. When$\nu$ has a density in $L\_\infty(\lambda)$ then we obtain two sequences of finite moments vectorsof increasing size (the number of moments) which converge to the moments of $\nu$ and $\psi$ respectively, as the number of moments increases. Importantly, {\it no} \`a priori knowledge on the supports of $\mu, \nu$ and $\psi$ is required.