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Consider a sequence of complex valued measures \mu_{n} in the euclidean space \R^d which converges weakly to some compactly supported measure \mu. The weak convergence is in the sens that \int_{\R^d} \psi d\mu_n converges to \int_{\R^d} \psi d\mu for each smooth function with compact support $\psi$. My problem is I want to know if there is a way to extend this convergence to polynomials knowing that polynomials are integrable with respect \mu_n for each n. |
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No; here's an easy counterexample. Let $\mu_n$ be the uniform measure on the interval $[n,n+1]$. This sequence of compactly supported measures converges weakly to the zero measure, in the sense you described, because the supports of the $\mu_n$'s eventually move away from the compact support of your $\psi$. Furthermore, all polynomials are certainly integrable with respect to each $\mu_n$, but those integrals don't tend to zero as $n\to\infty$ (except for the zero polynomial). |
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