free probability theory. Section 3 provides complete descriptions about the connections among free, Boolean convolutions, and $\boxplus$-infinitely divisible measures. Section 4 investigates the set of $\boxplus$-infinitely divisible measures with mean zero and finite variance. Section 5 contains results about the supports and regularity for the measures $\left(\mu^{\boxplus
p}\right)^{\uplus q}$, where $p\geq1$ and $q>0$.

Preliminary
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For any complex number $z$ in $\mathbb{C}$, let $\Re z$ and $\Im z$ be the real and imaginary parts of $z$, respectively. Denote by $\mathbb{C}^+=\{z\in\mathbb{C}:\Im z>0\}$ the complex upper half-plane. Consider the set $\mathcal{G}$ defined as $$\mathcal{G}=\left\{G|G:\mathbb{C}^+\to\mathbb{C}^-\;\;\mathrm{is}\;\;\mathrm{analytic}\;\;\mathrm{and}
\;\;\lim_{y\to\infty}iyG(iy)=1\right\}.$$ It is known that a function $G$ is in $\mathcal{G}$ if and only if there exists some measure $\mu\in\mathcal{M}$ such that $G$ can be written as $$G(z)=G_\mu(z):=\int_\mathbb{R}\frac{1}{z-s}\;d\mu(s),\;\;\;\;\;z\in\mathbb{C}^+.$$ The function $G_\mu$ is called the Cauchy transform of $\mu$. The measure $\mu$ can be recovered from $G_\mu$ as the weak limit of the measures $$\label{inversion}
d\mu_\epsilon(s)=-\frac{1}{\pi}\Im G_\mu(s+i\epsilon)\;ds$$ as $\epsilon\to0^+$. This is the Stieltjes inversion formula. Particularly, if $\Im G$ extends continuously to an open interval containing some point $x\in\mathbb{R}$ then the density of the absolutely continuous part of $\mu$ at $x$ is given by $-\Im
G(x)/\pi$.

Another class of functions which is closely related to $\mathcal{G}$ and plays a 