particularly, the compound free Poisson distribution with the rate $\lambda$ and jump distribution $\nu\in\mathcal{M}$ is shown to be of the form $(\nu_1^{\boxplus(\lambda+1)})^{\uplus\lambda/(\lambda+1)}$, $\nu_1\in\mathcal{M}$, and its $\boxplus$-divisibility indicator is calculated as well. In the study of the measures with mean zero and finite variance $\sigma^2$, we reformulate their $\boxplus$-divisibility indicators in terms of free Brownian motion: $$\mathrm{Ind}(\mu)=\sup\left\{t\geq0:E_\mu=\sigma^2G_{\nu_t\boxplus\gamma_{t\sigma^2}}\;\;
\mathrm{for}\;\;\mathrm{some}\;\;\nu_t\in\mathcal{M}\right\}.$$ As a consequence of this reformulation, a measure $\nu\in\mathcal{M}$ can be written as $\nu_1\boxplus\gamma_t$ for some $\nu_1\in\mathcal{M}$ and $t>0$ if and only if $\mathrm{Ind}(\Phi^{-1}(\nu))>0$. Moreover, we have $\mathrm{Ind}(\mu)>1$ if and only if $\phi_\mu=\sigma^2G_{\nu\boxplus\gamma_t}$ for some $\nu\in\mathcal{M}$ and $t>0$. The work \[4\] provides solid foundations for the current research and leads us to investigate the supports and regularity for the measures $\left(\mu^{\boxplus
p}\right)^{\uplus q}$, $p\geq1$, $q>0$. We prove that the nonatomic parts of this type of measure are absolutely continuous and the densities are analytic wherever they are positive. More importantly, the number of components in the support of $((\mu^{\boxplus
p})^{\uplus q})^{\mathrm{ac}}$ is independent of $q$ and a decreasing function of $p$. Particularly, $(\mu^{\uplus
q})^{\mathrm{ac}}$ contains the same number of components in the support for any $q>0$ provided that $\mathrm{Ind}(\mu)>0$.

The paper is organized as follows. Section 2 contains definitions and basic facts in 