= Local finiteness does not imply finite intensity
Here is an explicit example even within <simple point processes>. Let $K$ be a positive integer with <probability mass function> $\mathbb P(K=k)=1/[k(k+1)]$. The telescoping series sums to one, $K$ is finite <almost surely>, but $\mathbb E[K]=\sum_{k\ge1}1/(k+1)=\infty$. Conditional on $K$, sample $K$ <independent> points with the <uniform distribution> on $(0,1)$. The resulting <point process> has finitely many distinct points <almost surely>, while $\mu((0,1))=\infty$. Thus finite counts in each realization and finite mean counts are separate requirements; local finiteness of the <intensity measure> is an integrability assumption.
Back to article page
