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A stationary point process of finite intensity has mean measure rho times Lebesgue measure

Statement

Assume AC and Countable Choice, as carried by the random-point-process supplier. Let $η$ be a stationary locally finite point process on $R^{d}$ with $0 < ρ : = E [η (Q)] < \infty$ , where $Q = [0, 1)^{d}$ . Its mean measure $ν (B) : = E [η (B)]$ is

$ν (B) = ρ λ_{d} (B)$

for every Borel $B \subseteq R^{d}$ .

Facts & Assumptions

Given: AC, Countable Choice, a stationary point process $η$ , and $0 < ρ = E [η (Q)] < \infty$ .

[A1]

AC is carried by the transitive point-process supplier dependencies. (The Axiom of Choice)

[A2]

Countable Choice is carried by the transitive point-process supplier dependencies and is used for the Borel Lebesgue measure calculus. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

For a measurable random locally finite Borel measure, $ν (B) = E [η (B)]$ is a measure; all evaluations are measurable. (Mean measures and measurable nonnegative integrals of random measures)

[F2]

Stationarity means the law is invariant under $θ_{x}$ , and $ρ = E [η (Q)]$ for $Q = [0, 1)^{d}$ . (Stationarity, finite intensity, and spatial ergodicity)

[F3]

Every locally finite translation-invariant Borel measure $m$ on $R^{d}$ satisfies $m (B) = m (Q) λ_{d} (B)$ . (A locally finite translation-invariant Borel measure on $R^{d}$ is a multiple of Lebesgue measure)

Proof

Given: AC, Countable Choice, a stationary point process $η$ with $ρ = E η (Q) \in (0, \infty)$ , and its mean measure $ν$ .

1.1givenF1F2

By [F1], $ν (B) = E [η (B)]$ is a Borel measure, and $ν (Q) = ρ$ .

2.1step 1.1F2algebra

Let $B$ be bounded Borel. Choose an integer $n$ with $B \subseteq [- n, n)^{d}$ . This cube is a finite disjoint union of translates $Q + k$ , $k \in {- n, \dots, n - 1}^{d}$ . Stationarity [F2] gives $ν (Q + k) = E [η (Q + k)] = ρ$ for every such $k$ . Monotonicity and finite additivity therefore give $ν (B) \leq (2 n)^{d} ρ < \infty$ . Hence $ν$ is locally finite.

2.2step 1.1F2

For any Borel $B$ and translation $x$ , the definition of recentering gives $ν (B + x) = E [η (B + x)] = E [(θ_{x} η) (B)] = E [η (B)] = ν (B),$ where the third equality is stationarity [F2]. Thus $ν$ is translation invariant.

3.1A1A2step 1.1step 2.1step 2.2F3∎

Apply [F3] to the locally finite translation-invariant measure $ν$ . Since $ν (Q) = ρ$ by step 1.1, it follows that $ν (B) = ρ λ_{d} (B)$ for every Borel $B$ . This is the claimed mean-measure formula.

Sources

Last and Penrose, Chapter 8, §8.1 and Proposition 8.2; Baddeley, Lecture 2, §2.4. The proof checks the mean-measure hypotheses locally.

Depends on

Used by

Dependency tree · two levels

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Sources

Günter Last and Mathew Penrose, Lectures on the Poisson Process (2017 author-hosted version) (standard reference, not scraped)
Adrian Baddeley, Spatial Point Processes and their Applications (standard reference, not scraped)