On the moment problem of closed semi-algebraic sets

Let $K_f$ be a closed semi-algebraic set in $\dR^d$ such that there exist bounded real polynomials $h_1,{...},h_n$ 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 $C_\lambda=\{x:h_1(x)=\lambda_1,{...},h_n(x)=\lambda_n\}$ 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