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22), and because the number of sets A ∈ A and integers k < ∞ is countable, this requirement is satisfied almost surely. s. extended to a measure on σ(A). The necessity of both conditions follows directly from the additivity and continuity properties of a measure. s. 24b) 18 9. XV. s. finite-valued (finite integer-valued) on bounded Borel sets. 24b)] hold for all sequences {An } of bounded Borel sets with An ↓ ∅. Proof. IX. 22) hold for Borel sets in general, they certainly hold for sets in A. XIV are satisfied, and we can assert that with probability 1 the ξA (ω), initially defined for A ∈ A, can be extended to measures ξ ∗ (A, ω) defined for all A ∈ σ(A) = BX .

X , meaning a countable family T = {Tn } of ‘infinite’ dissecting systems of 16 9. 3 Tilings Tn = {Ani }, Tn+1 = {An+1,i } with sets containing x, y. X : each tiling Tn = {Ani : i = 1, 2, . } consists of countably many disjoint bounded sets Ani satisfying (i) (Partition and tiling properties) Ani ∩Anj = ∅ for i = j, and i Ani = X ; (ii) (Nesting property) An−1,i ∩ Anj = Anj or ∅, and any An−1,i ∈ Tn−1 is expressible An−1,i = j∈Jn−1,i Anj for some finite set of indices Jn−1,i ; (iii) (Point-separating property) given distinct x, y ∈ X , there exists an inte/ Ani ; and ger n(x, y) such that for n ≥ n(x, y), x ∈ Ani implies y ∈ (iv) (Enumeration consistency property) given distinct x, y ∈ X , when Tm is such that x ∈ Ami y, for all n ≥ m sets An,in x, An+1,in+1 x and An,jn y, An+1,jn+1 y are such that jn − in and jn+1 − in+1 have the same sign.

VI are not only necessary but also sufficient to ensure that the fidi distributions can be associated with a random measure. 1(f) implies that without nonnegativity, the sufficiency argument breaks down. It appears to be an open problem to find necessary and sufficient conditions on the fidi distributions that ensure that they belong to a random signed measure. 4. VII. V. VI. Proof. XIV, so we proceed to sufficiency. 1). 21) for each fixed pair of bounded Borel sets. s. 22) for each fixed sequence of bounded Borel sets An with An ↓ ∅.

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An Introduction to the Theory of Point Processes by GUJARATI

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