Change of Variables in an Integral
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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.
KeywordsIndependent Function Infinite Product Smooth Vector Field Stieltjes Measure Increase Distribution Function