Weak convergence from convergence of integrals of polynomials

Question

Let $\mu_n$ for $n\in \mathbb{N}$ and $\mu$ be probability measures on $\mathbb{C}^d$ with uniformly bounded support. Suppose that $$\int_{\mathbb{C}^d} f(z_1, \ldots, z_d) d\mu_n \to \int_{\mathbb{C}^d} f(z_1, \ldots, z_d) d\mu $$ for any $f$ which is a polynomial in $z_1, \ldots, z_d, \overline{z_1}, \ldots, \overline{z_d}$. Does it follows that $\mu_n \to \mu$ weakly? I'm especially looking for a reference for this fact.

Answer 1

I realized that this follows directly from the Stone-Weierstrass theorem. Let $X$ be a compact subset of $\mathbb{C}$ containing the support of $\mu$ and all $\mu_n$. The algebra of polynomials in $z_1,\ldots, z_d, \bar{z_1}, \ldots, \bar{z_d}$ is a unital *-subalgebra of $C(X)$ and separates points, so by Stone-Weierstrass it is dense in $C(X)$.