Abstract
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.
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Notes
- 1.
Francesco Bonaventura Cavalieri (1598–1647)—Italian mathematician.
- 2.
Wacłav Franciszek Sierpiński (1882–1969)—Polish mathematician.
- 3.
Leonida Tonelli (1885–1946)—Italian mathematician.
- 4.
Guido Fubini (1879–1943)—Italian mathematician.
- 5.
Archimedes (\('A\rho\chi\iota\mu\acute{\eta}\delta\eta\varsigma\), circa 287 – 212 BC)—Greek mathematician and inventor.
- 6.
Joseph Liouville (1809–1882)—French mathematician.
- 7.
Emilio Gagliardo (1930–2008)—Italian mathematician.
- 8.
Louis Nirenberg (born 1925)—American mathematician.
- 9.
Sergey L’vovich Sobolev (1908–1989)—Russian mathematician.
- 10.
This problem was proposed by A. Andzans in a slightly different formulation (see “Kvant”, 1990, No. 3, p. 27, Problem M1211). The authors are grateful to A.N. Petrov for drawing their attention to this result.
References
Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in Analysis. Holden-Day, San Francisco (1964). 5.2.2
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Makarov, B., Podkorytov, A. (2013). The Product Measure. In: Real Analysis: Measures, Integrals and Applications. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5122-7_5
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DOI: https://doi.org/10.1007/978-1-4471-5122-7_5
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