Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights

We prove in a direct fashion that a multidimensional probability measure μ is determinate if the higher-dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the trigonometric functions, are dense in all associated Lp-spaces for 1≤p<∞. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under μ. We give practical examples of such weights, based on their classification. As in the one-dimensional case, the results on determinacy of measures supported on \Rn lead to sufficient conditions for determinacy of measures supported in a positive convex cone, that is, the higher-dimensional analogue of determinacy in the sense of Stieltjes.