  The fibre theorem \cite{schm2003} for the moment problem on closed
semi-algebraic subsets of $\R^d$ is generalized to finitely generated real
unital algebras. A charaterization of moment functionals on the polynomial
algebra $\R[x_1,\dots,x_d]$ 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}.
