By Michele Audin

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**Additional info for Les Systemes Hamiltoniens Et Leur Integrabilite (French Edition)**

**Sample text**

In this way, we obtain the following measurability and uniqueness result (using analogous notation). 1. The mapping X : (Ω, A, P) → (Ns , Ns ) is a point process if and only if {X(C) = 0} is measurable for all C ∈ C. Let X, X be simple point processes in E. If P(X(C) = 0) = P(X (C) = 0) D for all C ∈ C, then X = X . Proof. The ﬁrst assertion follows from the fact that the mapping Z := supp X : (Ω, A, P) → F is measurable if and only if Z −1 (F C ) is measurable for all C ∈ C. 3, the distribution of X is uniquely determined by the probabilities P(X ∩C = ∅), C ∈ C.

Ck ), Ci ∈ C. Since the sets C0 , C1 , . . ,Ck , it must be shown that this deﬁnition is unambiguous. It is then possible to prove that A is a semialgebra generating B(F) and that P is σ-additive on A. By the measure extension theorem, P can be extended to a probability measure on B(F). This probability measure satisﬁes P(FC ) = T (C) for all C ∈ C. The ﬁrst part of the proof (consisting of three lemmas) is combinatorial in nature and does not use topological properties; therefore, we formulate it for a general set system with appropriate properties.

We emphasize that ∅ ∈ K in this book, which is convenient, but diﬀers from common usage. We denote by R the convex ring, whose elements are the ﬁnite unions of convex bodies, also called polyconvex sets. The extended convex ring is the system S := {F ∈ F : F ∩ K ∈ R for all K ∈ K}. The elements of S are the countable unions of convex bodies with the property that every compact set hits only ﬁnitely many of the bodies. Clearly, K ⊂ R ⊂ S ⊂ F and R ⊂ C ⊂ F, where each inclusion is strict. However, cl R = F; in fact, every element of F is the limit of a sequence of ﬁnite sets.