= Solution
On a <measurable space> $(E,\mathcal E)$ with a sigma-finite <measure> $\mu$, a <Poisson random measure> $N$ is a random counting measure such that $N(A)$ has the <Poisson distribution> with parameter $\mu(A)$ for every measurable $A$ of finite intensity, and the counts on finitely many pairwise disjoint measurable sets are <independent>. The sample measures are countably additive; repeated atoms are allowed if the intensity has atoms.
When $\mu(A)=\infty$, exhaustion by finite-intensity subsets gives $N(A)=\infty$ almost surely. Indeed their counts increase, and the <Poisson distribution> <probability> of a count at most any fixed integer tends to zero as the parameter tends to infinity. \b[<Independent> Poisson counts on disjoint sets, with means prescribed by $\mu$, characterize the <Poisson random measure>.]
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