On the moment problem of closed semi-algebraic sets

Let $K_f$ be a closed semi-algebraic set in ${\text{dR}}^d$ such that there exist bounded real polynomials $h_1,\mathrm{.}\mathrm{.}\mathrm{.},h_n$ on $K_f$. It is proved that the moment problem for $K_f$ is solvable provided it is for all sets $K_f\cap C_{\lambda }$, where $C_{\lambda }=\{x:h_1(x)={\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},h_n(x)={\lambda }_n\}$ and $\inf \{h_j(x);x\in K_f\}\le {\lambda }_j\le \sup \{h_j(x);x\in K_f\}$. New classes of non-compact closed semi-algebraic sets $K_f$ are found for which the moment problem is solvable. To appear in Journal fuer die Reine und Angewandte Mathematik