ABSTRACT
Higher dimensional real analysis is much more complex and fascinating than analysis on the real line. Thus we are certainly interested in doing measure theory in R 2 $ {\mathbb{R}}^{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math13_1.tif"/> , for instance. We could, if we wished, use the open sets in R 2 $ {\mathbb{R}}^{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math13_2.tif"/> to generate the σ‐algebra of Borel sets and proceed from there. But it is natural to think of R 2 = R × R $ {\mathbb{R}}^{2} = {\mathbb{R}} \times {\mathbb{R}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math13_3.tif"/> and to wonder whether the 1‐dimensional Lebesgue measure on each of the R $ {\mathbb{R}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math13_4.tif"/> factors can somehow be combined to produce a product measure on R 2 . $ {\mathbb{R}}^{2} . $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math13_5.tif"/>
