The Product Measure
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This chapter is devoted to the important notion of product measure. Besides the results related to the general case, we discuss the question of representing the Lebesgue measure in a multi-dimensional space as a product of measures and prove Cavalieri’s principle. We give several examples of application of the results obtained. In particular, we obtain a formula connecting the Euler and Γ functions and prove the Gagliardo–Nirenberg–Sobolev inequality.
In the last section, we introduce the notion of an infinite product of measures, which is important in probability theory.
KeywordsProduct Measure Gagliardo Nirenberg Sobolev Inequality Infinite-dimensional Cube Standard Extension Semiring
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