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Spaces, Kernels, and Disintegration

The purpose of this chapter is to introduce some underlying framework and machinery, to form a basis for our subsequent development of random measure theory. The chapter also contains some more technical results about differentiation and disintegration, needed only for special purposes. The impatient reader may acquire some general familiarity with the basic notions and terminology from the following introduction, and then return for further details and technical proofs when need arises. To ensure both sufficient generality and technical flexibility, we take the underlying space S to be an abstract Borel space, defined as a measurable space that is Borel isomorphic to a Borel set B on the real line R. In other words, we assume the existence of a 1 − 1, bi-measurable mapping between S and B. In Theorem 1.1 we prove that every Polish space S is Borel, which implies the same property for every Borel set in S. Recall that a topological space is said to be Polish if it admits a separable and complete metrization. The associated σ-field S is understood to be the one generated by the topology, known as the Borel σ-field. For technical reasons, we restrict our attention to locally finite measures on S. In the absence of any metric or topology on S, we then need to introduce a localizing structure, consisting of a ring Sˆ ⊂ S of bounded sets with suitable properties. In fact, it is enough to specify a sequence Sn ↑ S in S, such that a set is bounded iff it is contained in one of the sets Sn . For metric spaces S, we may choose the Sn to be concentric balls of radii n, and when S is lcscH (locally compact, second countable, and Hausdorff), we may choose Sˆ to consist of all relatively compact subsets of S. The space MS of all locally finite measures μ on S may be equipped with ˆ or the σ-field generated by all evaluation maps πB : μ → μB with B ∈ S, equivalently by all integration maps πf : μ → μf = f dμ, with f ≥ 0 a measurable function on S. The space MS is again Borel by Theorem 1.5, as is the subspace NS of all integer-valued measures on S. Every measure μ ∈ MS has an atomic decomposition μ=α+



βk δσk ,

(1)

k≤κ

in terms of a diffuse (non-atomic) measure α, some distinct atoms σk ∈ S, and some real weights βk > 0. Here δs denotes a unit mass at s, defined by © Springer International Publishing Switzerland 2017 O. Kallenberg, Random Measures, Theory and Applications, Probability Theory and Stochastic Modelling 77, DOI 10.1007/978-3-319-41598-7_1

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Random Measures, Theory and Applications

δs B = 1B (s), where 1B is the indicator function of the set B. The sum is unique up to the order of terms, and we can choose α and all the σk and βk to be measurable functions of μ. When μ ∈ NS , we have α = 0, and βk ∈ N = {1, 2, . . .} for all k. To prove such results, we may consider partitions of S into smaller and smaller subsets. To formalize the idea, we may introduce a dissection sysˆ such that tem, consisting of some nested partitions of S into subsets Inj ∈ S, every bounded set is covered