  We prove a solvability theorem for the Stieltjes moment problem on $R^d$
which is based on the multivariate Stieltjes condition $\sum_{n=1}^\infty
L(x_j^n)^{-1/(2n)}=+\infty$, $j=1,\dots,d.$ This result is applied to derive a
new solvability theorem for the moment problem on unbounded semi-algebraic
subsets of $R^d$.
