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• W1566144857 abstract "A probability measure ν on a product space X × Y is said to be bistochastic with respect to measures λ on X and μ on Y if the marginals π1(ν) and π2(μ) are exactly λ and μ. A solution is presented to a problem of Arveson about sets which are of measure zero for all such ν. If (X,F , λ) and (Y,G, μ) are probability spaces, we shall say that a probability measure σ on the product σ-algebra F⊗G is bistochastic (with respect to λ and μ), and write σ ∈ Bist(λ, μ) if σ has λ and μ as its marginal measures π1(σ) and π2(σ), that is to say, if π1(σ)(A) = σ(A × Y ) = λ(A) and π2(σ)(B) = σ(X × B) = μ(B) for all A ∈ F and B ∈ G. In the course of his study of operator algebras associated with commutative lattices of self-adjoint projections on Hilbert space, Arveson [1] investigated the following problem: Given probability spaces (X,F , λ) and (Y,G, μ), and a subset E of X × Y , when is E a null set for all σ ∈ Bist(λ, μ)? It is clear that a sufficient condition for this to be the case is that there exist F ∈ F , G ∈ G with λ(F ) = μ(G) = 0 and E ⊆ (F × Y ) ∪ (X × G); sets E of this type are said to be “marginally null”. Arveson showed that a converse to this statement is valid under certain circumstances: When X and Y are compact Hausdorff spaces and λ, μ are regular Borel probability measures, then a closed subset of X × Y which is a null set for all σ ∈ Bist(λ, μ) is necessarily marginally null [1, Theorem 1.4.2]. The same conclusion holds if (X,F) and (Y,G) are standard Borel spaces and E is the complement in X × Y of a countable union of Borel rectangles [1, Theorem 1.4.3]. We shall show that in both of these cases the result may be extended to a wider class of sets E. In fact, we shall give a more precise result, expressing in terms of the measures λ and μ the supremum of σ(E) taken over all bistochastic σ. This may be viewed as an extension of a theorem of Sudakov [7, Theorem 9]. We use fairly standard measure-theoretic terminology. A subset of a Cartesian product X × Y is said to be a rectangle if it has the form A× B with A ⊆ X and B ⊆ Y . The product σ-algebra F ⊗ G of σ-algebras F on X and G on Y is the σ-algebra generated by all rectangles A×B (A ∈ F , B ∈ G). If ν is a measure on a σ-algebra containing F ⊗ G the marginals are the measures defined on F and G by π1(ν)(A) = ν(A × Y ), π2(ν)(B) = ν(X × B). In the topological theory we are concerned only with finite positive measures defined on the Borel σ-algebra B(X) Received by the editors August 29, 1994. 1991 Mathematics Subject Classification. Primary 28A35; Secondary 28A12, 47D25. c ©1996 American Mathematical Society" @default.
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• W1566144857 title "On a measure-theoretic problem of Arveson" @default.
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• W1566144857 doi "https://doi.org/10.1090/s0002-9939-96-03076-6" @default.
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