= Homogeneous Poisson point process
A <homogeneous Poisson point process> on $\mathbb R^d$ has constant intensity $\lambda\geq0$, with <intensity measure> $\mu(A)=\lambda|A|$ for <Lebesgue measure> $|A|$. Counts on disjoint <measurable sets> are <independent>, and $N(A)$ has a <Poisson distribution> of mean $\lambda|A|$ when that volume is finite.
On the real line, an interval of length $\ell$ is empty with <probability> $e^{-\lambda\ell}$. Homogeneity alone is insufficient: the randomly shifted lattice $\mathbb Z+U$, with $U$ uniform on $[0,1)$, has translation-invariant intensity one but exactly one point in each half-open unit interval. It therefore fails the Poisson count law. Constant average density and <independent> random counts are different requirements.<Lectures on the Poisson Process by Günter Last and others>[Chapter 3, Poisson count law and constant-intensity specialization]{ref}
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