\[thm1\]. This result means that the set $\mathcal{P}_1$ is analogous to the set of probability measures with compact supports.

\[prop13\] Let $\mu = \nu + \lambda \in \mathcal{P}_1$ be a decomposition as in (\[decomp\]), where $\nu$ is supported on $[-R_1, R_1]$ and $\lambda$ is supported on $(-\infty,-R_2] \cup [R_2, \infty)$. Then there exists $C>0$ depending on $\nu,\lambda$ such that $|m_n(\mu)| \leq C R^n$ for all $n \geq 1$, where $R:=\max \{R_1, R_2\}$.

Let $p$ be the density function of $\lambda$. We have $$|m_n(\nu)| \leq \int_{-R_1} ^{R_1} |x|^n \nu(dx) \leq \nu(\operatorname*{\mathbb{R}}) R_1 ^n$$ and $$|m_n(\lambda)| \leq \int_{\gamma_{R_2}} |z|^n|p(z)||dz| \leq CR_2 ^{n+1}$$ for a constant $C>0$. This completes the proof.

In Section \[Cha\] we introduce a canonical constant $R_\mu$ of $\mu \in \mathcal{P}_1$. Then we can take the constant $R$ arbitrarily near to $R_\mu$: $R_\mu = \inf \{R \geq 0; \sup_{n \geq 1} \frac{|m_n(\mu)|}{R^n} < \infty \}$.

Let ${\mathcal{P}(n)}$, ${\mathcal{NC}(n)}$, ${\mathcal{I}(n)}$ and ${\mathcal{M}(n)}$ be respectively the sets of partitions, non-crossing partitions, interval partitions and monotone partitions of $\{1, \cdots, n \}$. Only the monotone partitions are equipped with order structure. The reader is referred to [@H-S; @N-S1] for details. For a partition $\pi =\{V_1, \cdots, V_{k}\}\in {\mathcal{P}(n)}$ and a sequence $\{ r_n 