= Randomly shifted lattice point process
{title2=$\Pi=\sum_{k\in\mathbb Z}\delta_{k+U}$}
Let $U$ have the <uniform distribution> on $[0,1)$ and place a point at every $k+U$, $k\in\mathbb Z$. This is a <simple point process>, locally finite, and a <stationary point process>: a <translation> only changes the phase modulo one, preserving its <uniform distribution>. For every <Borel set> $A$, the <Tonelli theorem> gives
$$
\mathbb E[\Pi(A)]=\sum_{k\in\mathbb Z}\int_0^1\mathbf1_A(k+u)\,du=|A|.
$$
Its consecutive points are exactly one unit apart, despite having the same <intensity measure> as a unit-rate <Poisson point process>. This is the one-dimensional translated-grid construction.<Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13-18, 2004/Spatial Point Processes and their Applications>[Section 2.1, Example 2.3, printed pp. 26-27: randomly translated grids have constant intensity]{ref}
Every half-open interval of length one contains exactly one point. For $A=[0,1/2)$ and $B=[1/2,1)$, $\Pi(A)+\Pi(B)=1$, each count has the <Bernoulli distribution> with parameter $1/2$, and
$$
\operatorname{Cov}(\Pi(A),\Pi(B))=0-\frac12\frac12=-\frac14.
$$
More generally, an interval of length $L=m+r$, with $m$ a <nonnegative integer> and $0\le r<1$, contains $m$ or $m+1$ points with <probabilities> $1-r$ and $r$. Its count <variance> is $r(1-r)$, while a unit-rate <Poisson point process> has count <variance> $L$.
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