A method of moments for semigroups \(S\) without zero, but satisfying \(S=S+S\) (Q1590059)

If \((\mu_i)\) is a sequence of measures on \(\mathbb{R}^k\) having moments \(s_n(\mu_i)= \int_{\mathbb{R}^k} x^nd\mu_i(x)\) of all orders \(n\in \mathbb{N}^k_0\) and if for each \(n\in\mathbb{N}^k_0\) the sequence \((s_n(\mu_i))_{i\in \mathbb{N}}\) converges to some \(t_n\in\mathbb{R}\) then some subsequence of \((\mu_i)\) converges weakly to a measure \(\mu\) with moments \(s_n(\mu)=t_n\) for \(n\in \mathbb{N}^k_0\). This indeterminate method of moments is extended to abelian semigroups satisfying \(S=S+S\), for a suitably chosen topology on the space of measures involved. As a corollary the so-called Haviland's criterion for moment sequences is likewise extended. It is astonishing how involved and difficult the proofs of results like these turn out to be. Deep insight is required in order to develop the necessary reasoning. The author has given one more example of very substantial work.