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Ak , Ak+1 )ψ(B) = ∆(Ak+1 )[∆(A1 , . . 12a) (k = 1, 2, . 12b) define a difference operator ∆(A) and its iterates acting on any set function ψ(·) for A, A1 , A2 , . . , B in a ring of sets on which ψ(·) is defined. 16) in the lemma below; the sign convention in its definition here is opposite that of Kurtz and Kallenberg. XI. For every integer k ≥ 1 and all Borel sets A1 , A2 , . . , B, ∆(A1 , . . , Ak )P0 (B) = P{N (Ai ) > 0 (i = 1, . . , k), N (B) = 0}. 13) Proof. For k = 1 we have P{N (A1 ) > 0, N (B) = 0} = P0 (B) − P0 (A1 ∪ B) = ∆(A1 )P0 (B).

We establish this by induction on the index k of the minimal family of disjoint sets generating the given fidi distribution. Suppose first that there are just two sets A1 , 30 9. Basic Theory of Random Measures and Point Processes A2 in this family. The new distributions defined by our construction are F2 (A1 , A1 ∪ A2 ), F2 (A2 , A1 ∪ A2 ), and F3 (A1 , A2 , A1 ∪ A2 ). Consistency with the original distributions F2 (A1 , A2 ), F1 (A1 ), and F1 (A2 ) is guaranteed by the construction and by the marginal consistency for distributions of disjoint sets.

K} = ∆(A1 , . . 18b) = ∆(A1 , . . , Aj−1 , Aj+1 , . . Ak ) ψ(Aj ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ k Pr{Z (Ai ) = 0 (all i)} = ψ i=1 Ai ); nonnegativity is ensured by (i), summation to unity by (ii), and marginal consistency reduces to ∆(A1 , . . , Ak+1 )ψ(B) + ∆(A1 , . . , Ak )ψ(B ∪ Ak+1 ) = ∆(A1 , . . , Ak )ψ(B). 2. 18) (with Pr replaced by P ). s. s. and, being the limit of an integer-valued sequence, is itself integervalued or infinite. 18b), we have P {ζn (A) = 0} = ψ(A) for all n, so P {N (A) = 0} = ψ(A) (all bounded A ∈ R).

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