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Products of a measurable intensity over finite configurations are measurable

Statement

Let $W$ and $C (W)$ be as in Finite-window spatial observations and configurations, and let $f : W \to [0, \infty)$ be finite-valued and Borel measurable. For $η = \sum_{i = 1}^{n} δ_{x_{i}} \in C (W)$ , counting repeated locations with their multiplicities and taking the empty product to be $1$ , define $G_{f} (η) : = \prod_{i = 1}^{n} f (x_{i}) .$ Then $G_{f} : C (W) \to [0, \infty)$ is measurable.

Facts & Assumptions

Given: The finite-window configuration space, a finite-valued Borel function $f$ , and the product $G_{f}$ defined in the Statement.

[F1]

The sigma-algebra on $C (W)$ is generated by the evaluation maps $η \mapsto η (B)$ for Borel $B \subseteq W$ ; each configuration is a finite sum of Dirac masses, with repetitions allowed. (Finite-window spatial observations and configurations)

[F2]

A nonnegative measurable function is the pointwise increasing limit of nonnegative simple measurable functions. (Every nonnegative measurable function is the increasing limit of simple measurable functions)

[F3]

If $u, v$ are nonnegative measurable functions, then $u v$ is measurable: for $a \geq 0$ , ${u v > a} = ⋃_{q \in Q, q > 0} {u > q} \cap {v > a / q},$ and for $a < 0$ the superlevel set is the whole space. Iterating this fact gives measurability of finite products. The identity uses the extended-product convention $0 \cdot (+ \infty) = 0$ . (Extended-real-valued measurable functions)

Proof

Given: $f$ is finite-valued and Borel measurable on $W$ .

1.1givenF1F3construct

First suppose $f$ is simple, with value $c_{j} \geq 0$ on each member $B_{j}$ of a finite Borel partition of $W$ . Then $G_{f} (η) = \prod_{j} c_{j}^{η (B_{j})}$ , using $0^{0} = 1$ ; every exponent is an evaluation map. For each fixed $c_{j}$ , the map $m \mapsto c_{j}^{m}$ on $N_{0}$ is measurable: the inverse image of any Borel set is a union of the countable level sets ${η (B_{j}) = m}$ . This also covers $c_{j} = 0$ with $0^{0} = 1$ . The finite product is measurable by [F3].

2.1givenstep 1.1F2algebra∎

For general $f$ , choose simple functions $s_{m} ↑ f$ by [F2]. If $η = \sum_{i = 1}^{n} δ_{x_{i}}$ , then $G_{s_{m}} (η) = \prod_{i = 1}^{n} s_{m} (x_{i}) ↑ \prod_{i = 1}^{n} f (x_{i}) = G_{f} (η)$ ; the limit holds because the product has finitely many nonnegative factors, and it also holds for the empty configuration. Thus $G_{f}$ is a pointwise limit of measurable functions from step 1.1 and is measurable.

Depends on

Used by

Finite-window spatial Poisson likelihood relative to unit rate Theorem

Dependency tree · two levels

24 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

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