Our goal in the next few lectures is to define point processes on the space of closed subsets of \(\mathbb{R}^d\).
Today:
Let \[ \mathcal{F} := \{\text{closed subsets of }\mathbb{R}^d\}, \ \mathcal{C} := \{\text{compact subsets of }\mathbb{R}^d\}, \ \mathcal{O} := \{\text{open subsets of }\mathbb{R}^d\}. \]
For \(A\subset\mathbb{R}^d\), define \[ \mathcal{F}^A := \{ F\in\mathcal{F} : F\cap A = \varnothing \}, \qquad \mathcal{F}_A := \{ F\in\mathcal{F} : F\cap A \neq \varnothing \}. \]
For subsets \(A_1,\ldots,A_k\subset\mathbb{R}^d\), define
\[\begin{align*} \mathcal{F}^A_{A_1,\ldots,A_k} &:= \mathcal{F}^A\cap \mathcal{F}_{A_1}\cap\cdots\cap \mathcal{F}_{A_k} \\ &= \{ F\in\mathcal{F} : F\cap A = \varnothing, F\cap A_i \neq \varnothing, \; \forall i \}. \end{align*}\]
Definition (Fell Topology). The Fell topology \(\mathcal{T}\) on \(\mathcal{F}\) is the topology generated by the base \[ \{ \mathcal{F}^C \cap \mathcal{F}_{G_1,\ldots,G_n} : C\in\mathcal{C}, \; G_1,\ldots,G_n\in\mathcal{O}, \; n\in\mathbb{N} \}. \]
Recall:
arbitrary unions of sets in \(\mathcal{T}\) are in \(\mathcal{T}\),
finite intersections of sets in \(\mathcal{T}\) are in \(\mathcal{T}\).
Elements of \(\mathcal{T}\) are called open sets.
A base for \(\mathcal{T}\) is a family of subsets of \(\mathcal{T}\) such that every open set is a union of base elements.
Remarks: The Fell topology is also generated by the collection of subsets
\[\begin{equation} \{ \mathcal{F}^C : C\in\mathcal{C} \} \; \cup \; \{ \mathcal{F}_G : G\in\mathcal{O} \}. \end{equation}\]
We will often consider the set \(\mathcal{F}' := \mathcal{F}\setminus\{\varnothing\}\) of non-empty closed setsets of \(\mathbb{R}^d\). \(\mathcal{F}'\) is equipped with the subspace topology \[ \mathcal{T}' := \{ A\cap \mathcal{F}' : A\in\mathcal{T} \}. \]
**Theorem (12.2.1 in [schneider2008stochastic).** ] \((\mathcal{F},\mathcal{T})\) is a compact, separable, Hausdorff topological space.
\((\mathcal{F}',\mathcal{T}')\) is a locally compact, separable, Hausdorff topological space.
To show compactness of \(\mathcal{F}\) one uses Alexander’s theorem, implying it suffices to show that every covering of \(\mathcal{F}\) by sets in the subbasis Eq.~[ref] contains a finite covering of \(\mathcal{F}\), and one can show this is not the case for the covering \(\bigcup_{D\in\mathcal{D}} \mathcal{F}_D\) of \(\mathcal{F}'\), where \(\mathcal{D}\) is a countable base of \(\mathbb{R}^d\).
By Urysohn’s metrization theorem, \((\mathcal{F},\mathcal{T})\) admits a metric \(d\) generating \(\mathcal{T}\), such that \((\mathcal{F},d)\) is a compact separable metric space.
On the set \(\mathcal{X}' :=\) the space of nonempty compact and convex subsets of \(\mathbb{R}^d\),
the topology induced by the Hausdorff metric and the subspace topology coincide.
For \(K \in \mathcal{X}'\), the support function of \(K\) is defined by \[ h_K(x) := \sup_{y \in K} \langle x, y \rangle, \qquad x \in \mathbb{R}^d. \]
Since \(h_K(\alpha x) = \alpha\, h_K(x)\) for all \(\alpha>0\), one may restrict to \(h_K(u)\) for \(u \in S^{d-1}\).
Then \(h_K(u)\) is the (signed) distance to the supporting hyperplane of \(K\) in direction \(u\).
For \(K, L \in \mathcal{X}'\), the Hausdorff distance between them is \[ d_H(K,L) := \sup_{u \in S^{d-1}} \big| h_K(u) - h_L(u) \big|. \]
Let \(\mathcal{B}(\mathcal{F})\) be the Borel \(\sigma\)-algebra generated by the Fell topology, i.e. the smallest \(\sigma\)-algebra containing all open sets of \(\mathcal{F}\). (We also let \(\mathcal{B}(\mathcal{F}') := \{ A\cap\mathcal{F}' : A\in\mathcal{B}(\mathcal{F}) \}.\))
Proposition. The \(\sigma\)-algebra \(\mathcal{B}(\mathcal{F})\) is generated by any of:
For proof, see Lemma 2.1.1 in [schneider2008stochastic].
Definition. Let \((\Omega,\mathcal{A},\mathbb{P})\) be a probability space.
A random closed set (RACS) \(Z\) in \(\mathbb{R}^d\) is a measurable element of \((\mathcal{F}, \mathcal{B}(\mathcal{F}))\), i.e. it is a mapping $ Z: $.
Example. If \(\Phi\) is a point process on \(\mathbb{R}^d\), then the support \[ \mathrm{Supp}(\Phi) := \{ x\in\mathbb{R}^d : \Phi(\{x\})>0 \} \] is a random closed set.