Proposition \[prop3\]; this implies that $G_{\mu_N^M}(z)$ converges to $G_{\mu_{a,b}}(z)$ uniformly in $|z|\geq 2R$. Thus we obtain the weak convergence $\mu_N^M \to \mu_{a,b}$ using Theorem 2.5 in [@Maa].

Characterization of the set $\mathcal{P}_1$ {#Cha}
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In this section we characterize the class $\mathcal{P}_1$ in terms of analytic properties of $G_\mu(z)$ and its reciprocal $F_\mu(z) = \frac{1}{G_\mu(z)}$, and moreover in terms of $F_\mu^{-1}$ and Fourier transforms $\mathcal{F}_\mu(z)$. Similarity between $\mathcal{P}_1$ and probability measures with compact supports will be understood from the characterizations; in particular, characterization by means of Fourier transforms results in a kind of Paley-Wiener theorem.

We recall that $\mu \in \mathcal{P}_1$ has a density of the form $\sum_{n=2}^\infty \frac{a_n}{x^n}$ for sufficiently large $|x|$. Let $R_\mu$ be defined by $$R_\mu = \inf \{R >0; \mu|_{|x|\geq R}(dx) = \sum_{n=2}^\infty \frac{a_n}{x^n} dx \}.$$ For instance, $R_{\mu_{0,b}} = b$. $\mu$ is compactly supported if and only if $a_n = 0$ for all $n$. If $\mu$ is compactly supported, then $R_\mu$ is equal to $\inf \{R>0, {\text{supp~}}\mu \subset [-R, R] \}$. We will prove that $R_\mu$ plays a role similar to the compactly supported case.

We introduce notation and terminology. $\Omega^{-1}$ denotes the set $\{z \in \operatorname*{\mathbb{C}}; \frac{1}{z} \in \Omega \} \subset \operatorname*{\mathbb{C}_{+}}$, where $\Omega$ 