Lebesgue decomposition in action via semidefinite relaxations

Abstract

Given all (finite) moments of two measures μ and λ on $\mathbb {R}^{n}$, we provide a numerical scheme to obtain the Lebesgue decomposition μ = ν + ψ with ν≪λ and ψ ⊥ λ. When ν has a density in $L_{\infty }(\lambda )$ then we obtain two sequences of finite moments vectors of increasing size (the number of moments) which converge to the moments of ν and ψ respectively, as the number of moments increases. Importantly, no à priori knowledge on the supports of μ,ν and ψ is required.