A characterization of moment sequences by extensions

The fibre theorem \cite{schm2003} for the moment problem on closed semi-algebraic subsets of ${\mathbb{R}}^d$ is generalized to finitely generated real unital algebras. A charaterization of moment functionals on the polynomial algebra $\mathbb{R}[x_1,\dots ,x_d]$ in terms of extensions is given. These results are applied to reprove basic results on the complex moment problem due to Stochel and Szafraniec \cite{stochelsz} and Bisgaard \cite{bisgaard}.