Homogeneous Poisson point process

A homogeneous Poisson point process on ${\mathbb{R}}^d$ has constant intensity $\lambda \ge 0$, 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.^[ref]