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