p}$ with respect to $\mu$, i.e., $$F_{\mu^{\boxplus
    p}}(z)=F_\mu(\omega_p(z)),\;\;\;\;\;z\in\mathbb{C}^+\cup\mathbb{R},$$ and consequently $$F_{\mu^{\boxplus
    p}}(H_p(z))=F_\mu(z),\;\;\;\;\;z\in\overline{\Omega_p}.$$]{}

Complete characterizations of the supports of $\mu^{\boxplus p}$ were given in \[\[Huang\]\]. Following the notations in Proposition \[prop2.1\], we give below the results needed in the current research.

\[Hthm\] For $\mu\in\mathcal{M}$, define the function $\psi_p:\mathbb{R}\to\mathbb{R}$ by $\psi_p(x)=H_p(x+if_p(x))$, $x\in\mathbb{R}$, and the set $V_p^+=\{x\in\mathbb{R}:f_p(x)>0\}$. Then the following statements are true.

1.  [The function $\psi_p$ is a homeomorphism on $\mathbb{R}$.]{}

2.  [The measure $(\mu^{\boxplus
    p})^{\mathrm{ac}}$ is concentrated on the set $\psi_p(\overline{V_p^+})$ with density $$\label{density}
    \frac{d(\mu^{\boxplus
    p})^{\mathrm{ac}}}{dx}(\psi_p(x))=\frac{(p-1)pf_p(x)}{\pi|px-\psi_p(x)+ipf_p(x)|},
    \;\;\;\;\;x\in V_p^+.$$ ]{}

3.  [The number of the components in the support of $\mu^{\boxplus
    p}$ is a decreasing function of $p$.]{}

The set of $\boxplus$-infinitely divisible measures in $\mathcal{M}$ is closed under weak convergence of probability measures. As shown in \[\[HV2\]\], a necessary and sufficient condition for $\mu$ to be $\boxplus$-infinitely divisible is that $\phi_\mu$ belong to $\mathcal{E}$.

The Boolean convolution introduced in \[\[Boolean\]\] was defined via the functions in $\mathcal{E}$. Given $\mu_1$ and $\mu_2$ in $\mathcal{M}$, the measure $\nu$ satisfying the relation $$E_\nu=E_{\mu_1}+E_{\mu_2}$$ is called the Boolean convolution of $\mu_1$ and $\mu_2$, and it is denoted $\mu_1\uplus\mu_2$. For $\mu\in\mathcal{M}$ and a positive integer $n$, the $n$-fold Boolean convolution $\mu\uplus\cdots\uplus\mu$ denoted by $\mu^{\uplus n}$ satisfies 