On coordinate-permutation-invariant signed Radon measures on Cartesian powers of compact Hausdorff spaces: a look at de Finetti’s theorem

We give two proofs of a representation theorem for those totally finite signed Radon measures on an infinite power K of a compact Hausdorff space K (with its product topology) with the property of invariance under permutation of coordinates, in terms of powers of Radon probability measures on K. This is essentially a version of de Finetti’s theorem on exchangeable sequences of random variables. Finally we will also use it to deduce an interesting measure algebra isomorphism. 1 Notation and definitions Before proceeding we fix some notation. If (Ω,Σ, μ) is a probability space and J is a set, we write (Ω ,Σ⊗J , μ⊗J) for the standard product of a J-indexed family of copies of (Ω, Σ, μ). Henceforth we will suppose J is infinite. If (Ω ,T, μ) is a measure space such that T and μ are invariant under permutation of coordinates, we will refer to this σ-algebra and measure as permutationinvariant. If L is a compact Hausdorff space we write C(L) for Banach space of real-valued continuous functions on L, CC(L) for the corresponding space of complex-valued continuous functions and B(L) for the Borel σ-algebra of L. For any set J we will write PJ for the power set of J and for any cardinal α we write [J ] (respectively, [J ]) for the subset of PJ containing those subsets of J of cardinality equal to (respectively, less than) α; the only cases we will need are α finite and countably infinite, where we will write ω in place of α. We also note that the set [J ] is naturally upwards directed by inclusion. For K a compact Hausdorff space and i ∈ J we denote by πi the i coordinate map K → K, and more generally for I ⊆ J we write πI for the canonical projection K → K given by taking the I-indexed coordinates (we will find throughout that the order in which these coordinates are taken is irrelevant due to permutation invariance, and so may be chosen arbitrarily in each case). For I ∈ [J ] we write AK,J,I for the subalgebra of C(K ) containing the finite sums of functions of the form ∏ i∈I fi ◦ πi, and set AK,J = ⋃ I∈[J]<ω AK,J,I . If J is infinite then an elementary application of the Stone-Weierstrass theorem shows that AK,J is norm-dense in C(K). Given a compact Hausdorff space L, the Riesz representation theorem justifies identifying C(L)∗ with the space of totally finite signed Radon measures on L in the natural way. We write Pr(L) for the subset of C(L)∗ containing the Radon probability measures, that is Pr(L) = { λ ∈ C(L)∗ : ‖λ‖ = 1, ∫ L χL dλ = 1 } . As is standard, this is a compact Hausdorff space with its vague topology (that induced by the weak∗-topology on C(L)∗), and we shall henceforth take it with this topology unless otherwise stated. If λ ∈ Pr(K) then we see that λ⊗J ∈ Pr(K ), that is, it is a Radon probability measure on K . Hence for any f ∈ C(K) we may consider the function Ψf : Pr(K) → R given by Ψf(λ) = ∫ KJ f d(λ⊗J). In fact Ψf is continuous on Pr(K), as may readily be seen using an approximation