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. 



extension of the convergence of a sequence of measures
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). 


