The compound Poisson distribution is defined as(see [Levy processes and infinitely divisible distributions](http://rads.stackoverflow.com/amzn/click/1107656494) page: 18):


Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a compound distribution $\mu$ on $\mathbb{R}$ is called a *compound Poisson* if its characteristic function is


$$
\Phi\_{\mu}(t) = e^{c(\Phi\_\sigma - 1)}
$$


My question is: Why we need the condition $\sigma(\{0\})=0$?


Similarly, in the book [Probability: a comprehensive course](http://rads.stackoverflow.com/amzn/click/144715360X) (page: 333) stated the following theorem: 


A probability measure $\mu$ on $\mathbb{R}$ is infinitely divisible if and only if there exists a sequence of finite measures $(\nu\_n)\_{n\in\mathbb{N}}$ on $\mathbb{R} - \{0\}$ such that $CPoi\_{\nu\_n} \rightarrow \mu(n\rightarrow\infty)$, where $CPoi\_{\nu\_n}$ denote the compound distribution with intensity $\nu\_n$.


My question is: Why $\nu\_n$ is defined on $\mathbb{R}-\{0\}$ other than $\mathbb{R}$?



The Levy measure of the compound Poisson distribution
This requirement is not necessary, but convenient. 


The measure $\sigma$ is the distribution of jumps. If $\sigma(\{0\})>0$, then the jump is $0$ with positive probability, meaning that there is in fact no jump. 


With the restriction $\sigma(\{0\})=0$, the parameters $\sigma$ and $c$ are uniquely determined by the distribution of process. But if we allow jumps to be equal to zero, we can increase intensity of jumps, at the same time allowing some of them to be zero, which would lead to the same process.


Another point is that when analyzing the behavior of the process you need to know when it changes value. With $\sigma(\{0\})>0$ it would be possible that the underlying counting measure is positive, however, there is no jump, since it is equal to zero. 


I hope this is enough to convince you that $\sigma(\{0\}) = 0$ is a convenient assumption.


**EDIT** I've just noticed Jean Duchon's answer. He is right, the last paragraph was some mess. I can't remember what I was going to say, so I removed it.


