Let X×Y denote the Cartesian product of two sets X and Y, which is the set of all ordered pairs (x, y) where x ∈ X and y ∈ Y. Consider two measure spaces (X, X, μ) and (Y, Y, η). In this section we construct a σ-algebra of subsets of the Cartesian product X×Y, denoted by X×Y, which is induced by the σ-algebras X and Y, such that a measure π on X×Y is given by the product of the measures μ on X and η on Y.

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Product Measures

  • Carlos S. Kubrusly

摘要

Let X×Y denote the Cartesian product of two sets X and Y, which is the set of all ordered pairs (x, y) where x ∈ X and y ∈ Y. Consider two measure spaces (X, X, μ) and (Y, Y, η). In this section we construct a σ-algebra of subsets of the Cartesian product X×Y, denoted by X×Y, which is induced by the σ-algebras X and Y, such that a measure π on X×Y is given by the product of the measures μ on X and η on Y.