Density questions in the classical theory of moments

Let μ be a positive Radon measure on the real line having moments of all orders. We prove that the set P of polynomials is note dense in L p ( R , μ ) for any p > 2 , if μ is indeterminate. If μ is determinate, then P is dense in L p ( R , μ ) for 1 ≤ p ≤ 2 , but not necessarily for p > 2 . The compact convex set of positive Radon measures with same moments as μ is studied in some details.