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Integration on Product Spaces

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Real and Abstract Analysis
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Abstract

Suppose that (X, ℳ, μ) and (Y, \( \mathcal{N} \), ν) are two measure spaces. We wish to define a product measure space

$$(X \times Y,{\mathcal{M}} \times {\mathcal{N}},\mu \times \nu),$$

where \({\mathcal{M}} \times {\mathcal{N}}\) is an appropriate σ-algebra of subsets of X × Y and μ × ν is a measure on \({\mathcal{M}} \times {\mathcal{N}}\) for which

$$\mu \times \nu (A \times B) = \mu (A) \cdot \nu (B)$$

whenever A ∈ ℳ and \(B \times {\mathcal{N}}\) That is, we wish to generalize the usual geometric notion of the area of a rectangle. We also wish it to be true that

$$\int\limits_{X \times Y} {fd\mu \times \nu = } \int\limits_X \int\limits_Y {fd\nu \ d\mu } = \int\limits_Y \int\limits_X {fd\mu \ d\nu },$$
((1))

for a reasonably large class of functions f on X × Y. Thus we want a generalization of the classical formula

$$\int\limits_{[a,b] \times [c,d]} {f(x,y) \ dS} = \int\limits_a^b \int\limits_c^d {f(x,y)} dy \ dx = \int\limits_c^d \int\limits_a^b {f(x,y)} dx \ dy,$$

which, as we know from elementary analysis, is valid for all functions \(f \in {\mathcal{S}}([a,b] \times, [c,d])\).

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© 1965 Springer-Verlag Berlin · Heidelberg

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Hewitt, E., Stromberg, K. (1965). Integration on Product Spaces. In: Real and Abstract Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88044-5_6

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  • DOI: https://doi.org/10.1007/978-3-642-88044-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-88046-9

  • Online ISBN: 978-3-642-88044-5

  • eBook Packages: Springer Book Archive

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