Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Point sets and certain classes of sets
- 2 Measures: general properties and extension
- 3 Measurable functions and transformations
- 4 The integral
- 5 Absolute continuity and related topics
- 6 Convergence of measurable functions, Lp-spaces
- 7 Product spaces
- 8 Integrating complex functions, Fourier theory and related topics
- 9 Foundations of probability
- 10 Independence
- 11 Convergence and related topics 223
- 12 Characteristic functions and central limit theorems
- 13 Conditioning
- 14 Martingales
- 15 Basic structure of stochastic processes
- References
- Index
7 - Product spaces
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Point sets and certain classes of sets
- 2 Measures: general properties and extension
- 3 Measurable functions and transformations
- 4 The integral
- 5 Absolute continuity and related topics
- 6 Convergence of measurable functions, Lp-spaces
- 7 Product spaces
- 8 Integrating complex functions, Fourier theory and related topics
- 9 Foundations of probability
- 10 Independence
- 11 Convergence and related topics 223
- 12 Characteristic functions and central limit theorems
- 13 Conditioning
- 14 Martingales
- 15 Basic structure of stochastic processes
- References
- Index
Summary
Measurability in Cartesian products
Up to this point, our attention has focussed on just one fixed space X. Consider now two (later more than two) such spaces X, Y, and their Cartesian product X × Y, defined to be the set of all ordered pairs (x, y) with x ∈ X, y ∈ Y. The most familiar example is, of course, the Euclidean plane where X and Y are both (copies of) the real line ℝ.
Our main interest will be in defining a natural measure-theoretic structure in X × Y (i.e. a σ-field and a measure) in the case where both X and Y are measure spaces. However, for slightly more generality it is useful to first consider σ-rings S, T in X, Y, respectively and define a natural “product” σ-ring in X ×; Y.
First, a rectangle in X ×; Y (with sides A ⊂ X, B ⊂ Y) is defined to be a set of the form A × B = {(x, y): x ∈ A, y ∈ B}. Rectangles may be regarded as the simplest subsets of X ×; Y and have the following property.
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- Chapter
- Information
- A Basic Course in Measure and ProbabilityTheory for Applications, pp. 141 - 176Publisher: Cambridge University PressPrint publication year: 2014