Converging to moments obeying Carleman's condition I believe that the following is true, and I'd like to make sure that it is and to have a reference.  Suppose that $\mu_N$ are a sequence of measures on $\mathbb{R}$.  Let $m_{N,k}$ be the $k$-th moment of $\mu_N$.  Suppose that, for each $k$, $m_{N,k}$ converges to a limit $m_k$ as $N\rightarrow \infty$; no uniformity assumption is made, in that for each $k$ the convergence may be different.  Suppose further that $m_k$ obeys Carleman's condition:
$\sum_{n=1}^\infty m_{2n}^{-\frac{1}{2n}}=\infty$
Then, I would like to know that the following 3 results are true:
1) There exists a measure $\mu$ such that the moments of $\mu$ are equal to $m_k$.
2) Such a measure $\mu$ is unique.
3) The measures $\mu_N$ weakly converge to $\mu$ as $N \rightarrow \infty$. 
Result (2) I think follows straightforwardly from Carleman's, but I want to verify that (1),(3) are correct.
 A: Yes, this works. It's convenient to view the $\mu_n$ as measures on the compact space $\mathbb R_{\infty}$, with $\mu_n(\{\infty\})=0$. By a diagonal process, we then find a subsequence (which I'll write as the original sequence) such that $x^{2k}\, d\mu_n(x) \to d\rho_k(x)$ weak $*$ for certain measures $\rho_k$ for all $k$; this follows because $\int_{\mathbb R_{\infty}} x^{2k}\, d\mu_n(x)\le C_k$ by assumption.
From this condition, we also deduce that $\rho_k(\{\infty\})=0$.
By comparing integrals of compactly supported continuous functions, we then find that $d\rho_k(x) = x^{2k}\, d\rho_0(x)$.
This measure $\rho=\rho_0$ has the correct even moments, by construction, and if $n$ is odd, then we write
$$
\int x^n\, d\rho(x) = \int x^{n-1} g(x)\, d\rho(x) + \int x^{n+1} h(x)\, d\rho(x),
$$
with both $g,h\in C(\mathbb R_{\infty})$, and $g$ supported near $x=0$ and $h$ supported away from $x=0$. Again, it follows that $\int x^n\, d\rho(x)= m_n$.
This gives (1). Then (2) is clear because the moment problem with the $m_n$'s is determinate, and then (3) also follows because, as we saw, the only possible limit point of the $\mu_n$ is the unique measure $\rho$ with these moments.
