Abstract
Suppose that (X, ℳ, μ) and (Y, \( \mathcal{N} \), ν) are two measure spaces. We wish to define a product measure space
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
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
for a reasonably large class of functions f on X × Y. Thus we want a generalization of the classical formula
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
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