Let $P$ and $Q$ be discrete probability mass' defined on countable subsets $\Omega_P$ and $\Omega_Q$ on $\mathbb{R}$.
A collection of $(P_t)_{t\in I}$ of probability mass' is called a Convolution Semigroup if for all $s,t\in I$ the identity $P_t \star P_s= P_{t+s} $ is fulfilled.
A probability mass $P$ is called infinity divisible if for all $n\in\mathbb{N}$ there exists a probability mass $ Q_n$ with $$ P = Q_n \star ... \star Q_n. $$ (note in the last equation the operation is executed n times)
(i) Using the convolution theorem, show that the collection/family of $(Poi_\alpha)_{\alpha>0}$ is a infinitely divisible convolution semigroup. (where Poi refers to the Poisson distribution)
(ii) Show that there exists a unique\clear smallest Set $I\subseteq\mathbb{R}_+$ with $I\in\mathbb{N}_0$, so that the collection $(\delta_x)_{x\in I}$ of Dirac Mass' is an infinitely divisible convolution semigroup.
I know how to show that the Poisson distribution forms an infinitely divisible convolution semigroup using the characteristic function, but sadly I am struggling showing this using the convolution theorem. For the second question I am not too familiar with mass theory, so any help or solutions would be very much appreciated.