  We prove in a direct fashion that a multidimensional probability measure 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 the associated L_p spaces for all
finite p. In particular these three statements hold if the reciprocal of a
quasi-analytic weight has finite integral under the measure. 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 R^n lead to sufficient conditions for determinacy of measures
supported in a positive convex cone, i.e. the higher dimensional analogue of
determinacy in the sense of Stieltjes.
