Abstract

We show that the existence of disintegration for cylindrical measures follows from a general disintegration theorem for countably additive measures. 1 Notation A probability space (X, A, P) is a nonempty set X together with a sigma-algebra A on X and a probability P on A (that is, a nonnegative countably additive measure with PX = 1). When C is a set of subsets of a given set, σ(C) is the smallest sigma-algebra that contains C. When A and B are sigma-algebras on X and Y, respectively, σ(A ⊗ B) is the smallest sigma-algebra making the canonical projections πX: X × Y → X and πY: X × Y → Y measurable. When (X, A, P) and (Y, B, Q) are two probability spaces, a probability S on σ(A ⊗ B) is a joint probability if πX[S] = P and πY [S] = Q. A lattice on X is a class of subsets of X that is closed under finite unions and finite intersections. A class K of sets is semicompact if every countable class K0 ⊆ K such that K0 = ∅ contains a finite class K00 ⊆ K0 such that ⋂ K00 = ∅. ∗ Extracted from the author’s manuscript dated July 20, 1979. 1 Let (X, A, P) be a probability space and K ⊆ A. We say that K approximates P if for every E ∈ A and ε> 0 there is a K ∈ K such that K ⊆ E and P(E \ K) < ε. All linear spaces are assumed to be over the field R of reals. When Y and Z are locally convex spaces, L(Y, Z) is the set of continuous linear mappings from Y to Z. When I and J are two index sets such that J ⊆ I, the canonical projection from R I onto R J is denoted pIJ. When I is an infinite index set, let C(I) be the set of all subsets of R I of the form p −1 IJ (C) where J is finite, ∅ ̸ = J ⊆ I and C is a compact subset of RJ. Then C(I)δ is the set of all countable intersections of sets in C(I). 2 Disintegration theorem The following theorem is an immediate consequence of Theorem 3.5 in [4]. Theorem 1 Let (X, A, P) and (Y, B, Q) be two probability spaces, and let S be a joint probability on σ(A ⊗ B). Let Q be complete, and let K be a semicompact lattice closed under countable intersections and such that A = σ(K) and K approximates P. Then there exists a family of probabilities {Py}y∈Y on A such that (a) for every E ∈ A, the function y ↦ → PyE is B-measurable; (b) for every E ∈ A and F ∈ B we have S(E × F) = PyE dQ(y).

Citations

44	Radon measures on arbitrary topological spaces and cylindrical measures - Schwartz - 1973
6	Disintegration and compact measures - Pachl - 1978
4	On compact measures - Marczewski - 1953
1	Équations linéaires á coefficients aléatoires. Symposia Mathematica Vol. VII - Krée - 1971