DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

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Recentering a point configuration at a spatial location

Definition

Let $d \geq 1$ be an integer and let $μ$ be a locally finite counting measure on $R^{d}$ as in Random measures and locally finite point processes, with $S = R^{d}$ . For $x \in R^{d}$ , its recentering at $x$ is the counting measure $θ_{x} μ$ defined on Borel $B \subseteq R^{d}$ by

$(θ_{x} μ) (B) : = μ (B + x) .$

Thus an occurrence at $x$ is moved to $0$ , and in an occurrence representation $μ = \sum_{i} δ_{X_{i}}$ one has $θ_{x} μ = \sum_{i} δ_{X_{i} - x}$ . The action obeys $θ_{x + y} = θ_{x} \circ θ_{y}$ and $θ_{0}$ is the identity. Joint measurability in $(x, μ)$ is proved in The spatial recentering action is jointly measurable ↗.

Axiom convention. The point-process supplier used for this definition has transitive dependencies on AC and Countable Choice; the recentering formula itself makes no selections.

Depends on

Used by

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)