Date of Award
Spring 5-13-2016
Level of Access Assigned by Author
Campus-Only Thesis
Degree Name
Master of Arts (MA)
Department
Mathematics
Advisor
David Bradley
Second Committee Member
Andre Khalil
Third Committee Member
Thomas Bellsky
Abstract
The integral accounts for one half of the calculus first described by Isaac Newton. Its conception is humble; it is the inverse of the derivative. Newton’s definition lacks rigor and is not constructive. Famous mathematicians, including A. L. Cauchy, B. Riemann, and J. Darboux, attempted to define the integral in a rigorous way and to develop constructive theories in which the integral can handle larger classes of functions. H. Lebesgue was able to extend the Riemann integral to the class of measurable functions. However, Lebesgue’s theory could not recover a function from its derivative without assuming the integrability of the derivative.
Several nonabsolute extensions of Lebesgue’s theory were proposed in the twentieth century; while all are vastly different in their methods, they are all equivalent. A. Denjoy’s theory came first in 1912. It is the only constructive extension of the Lebesgue integral; however, the process cannot be practically carried out. O. Perron’s theory of integration, published in 1914, while simpler than Denjoy’s, is neither constructive nor practical. The extension of the Lebesgue integral defined by J. Kurzweil and R. Henstock in the 1960s is the latest major development in the endeavor to retrieve a function from its derivative. It is not constructive, but its notation is similar to Riemann’s.
Further developments in this area include the Khintchine integral, the ultrafilter integral, and the controlled Newton integral.
Recommended Citation
David, Danielle E., "Nonabsolute Extensions of the Lebesgue Integral on the Real Line" (2016). Electronic Theses and Dissertations. 2431.
https://digitalcommons.library.umaine.edu/etd/2431