  The paper is a sequel to the paper "Application of localization to the
multivariate moment problem" by the same author. A new criterion is presented
for a positive semidefinite linear functional on the real polynomial algebra to
correspond to a positive Borel measure on real n-space. The criterion is
stronger than Nussbaum's criterion and is similar in nature to a criterion of
Schmudgen. It is also explained how the criterion allows one to understand the
support of the associated measure in terms of the non-negativity of the linear
functional on a quadratic module of the real polynomial algebra. This latter
result extends a result of Lasserre. The techniques employed are the same
localization techniques employed already in two earlier papers by the same
author.
