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. This in turn leads to a sufficient integral condition for the… CONTINUE READING