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Change of Variables in an Integral

  • Boris Makarov
  • Anatolii Podkorytov
Chapter
  • 4.4k Downloads
Part of the Universitext book series (UTX)

Abstract

This chapter expresses one of the vital sides of the book, the explication of basic notions of measure theory in close connection with classical analysis. Sect. 6.1 introduces the notion of weighted image of a measure. A number of important particular cases is then explored based on this notion. They are: change of variables in the Lebesgue integral under a diffeomorphism, the calculation of integrals using the distributions on the line corresponding to random variables (the formula for expectation, etc.) and the calculation of integrals with respect to the surface area. The latter topic is also researched in Sect. 6.5 and Chap.  8).

The last two paragraphs are demonstrating the power of the developed methods using the proof of a number of important geometrical results as an example. In particular, we prove the theorem on smooth vector fields on a sphere (the hairy ball theorem), the retraction theorem, Brouwer’s fixed point theorem, and Ball’s inequality estimating the area of a cross section of a multi-dimensional cube by a plane.

Keywords

Independent Function Infinite Product Smooth Vector Field Stieltjes Measure Increase Distribution Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). 6.2.1, 8.4.2, 8.4.5 zbMATHGoogle Scholar
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    Koldobsky, A.: Fourier Analysis in Convex Geometry. Am. Math. Soc., Rhode Island (2005). 6.7.1 zbMATHGoogle Scholar
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    Milnor, J.: Analytic proofs of the “hairy ball theorem” and the Brouwer fixed point theorem. Am. Math. Mon. 85(7), 521–524 (1978). 6.6.1 MathSciNetzbMATHCrossRefGoogle Scholar
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    Rogers, C.A.: A less strange version of Milnor’s proof of Brouwer’s fixed-point theorem. Am. Math. Mon. 87(7), 525–527 (1980). 6.6.1 zbMATHCrossRefGoogle Scholar
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    Rogers, C.A., Shephard, G.C.: The difference body of a convex body. Arch. Math. 8, 220–233 (1957). 6.4.2 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Boris Makarov
    • 1
  • Anatolii Podkorytov
    • 1
  1. 1.Mathematics and Mechanics FacultySt Petersburg State UniversitySt PetersburgRussia

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