Date of Award
College of Science
Type of Degree
When considering the limit of a sequence of functions, some properties still hold under the limit, but some do not. Unfortunately, the integration is among those not holding. There are only certain classes of functions that still hold the integrability, and the values of the integrals under the limiting process. Starting with Riemann integrals, the limiting integrations are restricted not only on the class of functions, but also on the set on which the integral is taken. By redefining the integration process, Lebesgue integrals successfully and significantly extend both the class of the functions integrated and the sets on which the integral is taken. But the Lebesgue integral still does not hold under the limit even with some simply defined functions. We will attempt to solve this problem by defining a special type of measure to handle the limiting case and introducing an extension the set of real numbers.
Ly, Tue Ngoc, "Limiting Problems in integration and An Extension of The Real Numbers System" (2008). Theses, Dissertations and Capstones. 714.