Nonabsolute Extensions of the Lebesgue Integral on the Real Line

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