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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]

A Poisson point process is a point process with independent Poisson distribution counts on disjoint measurable sets. Its intensity measure $\mu$ specifies those counts: for $\mu (A)<\infty$,Thus both the expected value and variance of $N(A)$ are $\mu (A)$. The independence requirement concerns disjoint test sets; overlapping sets can share points and have dependent counts.^[ref]

For a homogeneous Poisson point process in Euclidean space, $\mu (A)=\lambda |A|$, with constant intensity $\lambda$ and volume $|A|$. For example, intensity $2$ on the real line gives expected count $2$ and empty-interval probability $e^{-2}$ in a unit interval. Counts in two disjoint unit intervals are independent.

The counting measure formulation records multiplicity. A non-atomic intensity measure gives a simple point process, with no coincident points; an atom of a measure can instead carry a random multiplicity. Poisson random measures make that counting formulation explicit, while Poisson thinning theorem and the Superposition theorem for Poisson point processes explain useful ways of constructing related processes.