The addition property of the Poisson distribution asserts that independent Poisson random variables add to a Poisson: if $N_k \sim \operatorname{Po}(\lambda_k)$ are independent and $\sum_k \lambda_k < \infty$, then $\sum_k N_k \sim \operatorname{Po}(\sum_k \lambda_k)$. This is the fundamental closure property of the Poisson family under convolution, and it extends from finite sums (where the proof is a direct generating function calculation) to countably infinite sums (where the Monotone Convergence Theorem justifies the passage to the limit). The result reflects a deep structural property of the Poisson distribution: it is infinitely divisible. Any $\operatorname{Po}(\lambda)$ random variable can be decomposed as a sum of arbitrarily many independent Poisson random variables with smaller parameters, and conversely, any sum of independent Poissons is Poisson. This closure property is what makes the Poisson distribution the natural counting distribution for random point processes: when counting events from independent sources, the total count remains Poisson. The addition property is the foundational ingredient in the construction of the [Poisson Random Measure](/theorems/1194). Countable additivity of the random measure $M$ — the requirement that $M(\bigcup_k A_k) = \sum_k M(A_k)$ for disjoint sets — is established by verifying that the sum of independent Poisson random variables is Poisson with the correct parameter. It also connects to the [Splitting Property](/theorems/1193): if a $\operatorname{Po}(\lambda)$ count is split into categories, the resulting counts are independent Poisson, and adding them back recovers the original $\operatorname{Po}(\lambda)$ by the present result.