= Intensity measure of a point process
{title2=$\mu(A)=\mathbb E\Pi(A)$}
= Mean measure of a point process
{synonym}
= Intensity measure
{synonym}
The <intensity measure of a point process> $\Pi$ is its expected counting <measure>:
$$
\mu(A)=\mathbb E[\Pi(A)],\qquad A\in\mathcal B(E).
$$
It may take the value $+\infty$.<Lectures on the Poisson Process by Günter Last and others>[Section 2.2, Definition 2.5, printed p. 12: intensity as the expected counting measure]{ref} Countable additivity follows by applying the <monotone convergence theorem> to the counts of finite unions of pairwise <disjoint observation windows>. If $\mu$ is locally finite and <absolutely continuous with respect to> <Lebesgue measure>, its <Radon-Nikodym derivative> is the <intensity function of a point process> $\lambda$:
$$
\mu(A)=\int_A\lambda(x)\,dx.
$$
This is an expected number per unit volume, not a <probability density function> for a randomly selected point. For example, a single deterministic point has $\mu=\delta_0$, a <Dirac measure>, and no <Lebesgue measure> density. The density representation requires its own assumption.<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, Definitions 2.1-2.2, printed pp. 26-27: expected counts and intensity functions]{ref}
For a bounded <Borel set> $W$, $\mathbb E[N_W]=\mu(W)$, but local finiteness alone does not ensure that this <expected value> is finite. Conversely, $\mu(W)<\infty$ implies $N_W<\infty$ <almost surely>, even when $W$ is unbounded. A constant finite <intensity function of a point process> on all of $\mathbb R^d$ has infinite total <intensity measure>, although every bounded window has finite <expected value>.
Back to article page
