We prove in a direct fashion that a multidimensional probability measure $\mu$ 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 $L_p$-spaces for $1\leq p<\infty$. In particular these three statements hold if the reciprocal of a quasi-analytic weight has finite integral under $\mu$. 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.
"Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights." Ann. Probab. 31 (3) 1205 - 1227, July 2003. https://doi.org/10.1214/aop/1055425776