Convergence of Lasserre’s hierarchy: the general case

Lasserre's moment-SOS hierarchy consists of approximating instances the generalized moment problem (GMP) with relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming (SDP) problems. Due generality initial GMP, applications this technology are countless, one can cite among them polynomial optimization (POP), optimal control (OCP), volume computation problem, stability sets approximation problems, solving nonlinear partial differential equations (PDE). The solution original GMP is then approximated finite truncatures its sequence. For each application, proving convergence these towards sequence gives valuable insight on including relaxed values GMP's value. This note proposes a general proof such convergence, regardless faced with, under simple standard assumptions. As byproduct proof, also obtains strong duality properties both in infinite dimensional relaxations.