On a Problem of Kingman

P. J. Fitzsimmons

Let (E, E) be a Lusin metrizable space and let Φ be a random measure on (E, E). More precisely, Φ is a random variable, defined on some probability space (Ω, F, P), with values in the (positive, σ-additive) measures on (E, E). We assume that Φ is completely random [3], in the sense that {Φ(A_{n}) : n ∈ N} is an independent sequence of random variables whenever {A_{n} : n ∈ N} is a collection of pairwise disjoint E-measurable sets.

It is shown in [3], under a mild σ-finiteness condition, that such a random measure Φ can be decomposed as

(1)  Φ = Φ_{f} + Φ_{d} + Φ_{0},

where Φ_{f} is a purely atomic measure with atoms of independent sizes located at the points of a deterministic subset of E, and Φ_{d} is a non-atomic deterministic measure. The remaining component Φ_{0} is the most interesting of the three, and is the subject of most of the discussion in [3]. In particular, it is shown there that Φ_{0} is equal in distribution to a purely atomic measure Φ_{∗}. In [4] an argument due to D. Blackwell [1] is adapted to prove that Φ_{0} is itself purely atomic (with probability 1), under a broad additional condition. Our aim in this note is to point out that there is a simple direct argument showing that Φ_{0} is purely atomic in general.

For simplicity, we assume in what follows that P[Φ(E) < ∞] = 1. This finiteness condition is stronger than the σ-finiteness condition mentioned earlier; in particular, the decomposition (1) is valid.

Let M denote the class of finite measures on (E, E), and let M_{a} denote the subclass of purely atomic measures. Let M be the σ-algebra on M generated by the maps μ ↦ μ(B), B ∈ E. In saying that Φ_{0} and Φ_{∗} have the same distribution, we mean that (Φ_{0}(B_{1}), Φ_{0}(B_{2}), …, Φ_{0}(B_{n})) has the same distribution as (Φ_{∗}(B_{1}), Φ_{∗}(B_{2}), …, Φ_{∗}(B_{n})) for all n-tuples (B_{1}, B_{2}, …, B_{n}) of elements of E and all n ∈ N. It then follows from the monotone class theorem that

(2)  P[Φ_{0} ∈ C] = P[Φ_{∗} ∈ C],  ∀C ∈ M.

In view of (2), we need only verify that M_{a} is M-measurable. But this is an immediate consequence of Lemma 2.3 on page 20 of [2]. Indeed, each μ ∈ M admits a unique decomposition μ = μ_{d} + μ_{a} into diffuse and purely atomic parts, and the mapping μ ↦ μ_{d} is M-measurable. Thus M_{a} = {μ ∈ M : μ_{d}(B) = 0, ∀B ∈ E} is M-measurable because E is countably generated.

References

[1] Blackwell, D.: Discreteness of Ferguson selections, Ann. Statist. 1 (1973) 356–358.

[2] Kallenberg, O.: Random Measures (Third edition). Akademie-Verlag, Berlin, 1983.

[3] Kingman, J. F. C.: Completely random measures, Pacific J. Math., 21 (1967) 59–78.

[4] Kingman, J. F. C.: Poisson Processes, Oxford University Press, New York, 1993.