The paper is a sequel to the paper "Application of localization to themultivariate moment problem" by the same author. A new criterion is presentedfor a positive semidefinite linear functional on the real polynomial algebra tocorrespond to a positive Borel measure on real n-space. The criterion isstronger than Nussbaum's criterion and is similar in nature to a criterion ofSchmudgen. It is also explained how the criterion allows one to understand thesupport of the associated measure in terms of the non-negativity of the linearfunctional on a quadratic module of the real polynomial algebra. This latterresult extends a result of Lasserre. The techniques employed are the samelocalization techniques employed already in two earlier papers by the sameauthor.
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