Let V be the countable dimensional real polynomial algebra. Let \tau be a locally convex topology on V. Let K be a closed subset of Rⁿ, and let M := M{g₁,..., g_s} be a finitely generated quadratic module in V. We investigate the following question:

When is the cone Psd(K) (of polynomials nonnegative on K) included in the \tau closure of M?

We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of the cone of sums of squares with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) for certain convex compact K. We use these results to generalize Berg et al. work on exponentially bounded moment sequences. (Joint work with Mehdi Ghasemi and Ebrahim Samei)