Gauge integration
02Sep_McInnis.pdf (288.2Kb)Abstract
It is generally accepted that the Riemann integral is more useful as a pedagogical device for introductory analysis than for advanced mathematics. This is simply because there are many meaningful functions that are not Riemann integrable, and the theory of Riemann integration does not contain su.ciently strong convergence theorems. Lebesgue developed his theory of measure and integration to address these shortcomings. His integral is more powerful in the sense that it integrates more functions and possesses more general convergence theorems. However, his techniques are signi.cantly more complicated and require a considerable foundation in measure theory. There is now an impetus to accept the gauge integral as a possible new standard in mathematics. This relatively recent integral possesses the intuitive description of the Riemann integral, with the power of the Lebesgue integral. The purpose of this thesis is to explore the basis of gauge integration theory through its associated preliminary convergence theorems, and to contrast it with other integration techniques through explicit examples.
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
Related items
Showing items related by title, author, creator and subject.
-
Riemann-Stieltjes optimal control problems for uncertain dynamic systems
(AIAA, 2015);Motivated by uncertain parameters in nonlinear dynamic systems, we define a nonclassical optimal control problem where the cost functional is given by a Riemann–Stieltjes “functional of a functional.” Using the properties ... -
Some consequences of a covariant differential coordinate system
(Monterey, California: U.S. Naval Postgraduate School, 1961);The assumption is made that the concept of a covariant coordinate system is meaningful and that such a system must satisfy the differential equations a) dx(I) = g(ij)dx (j) where the g(ij) are the components of the ... -
Consistent approximation of an optimal search problem
(IEEE, 2012-12-10);This paper focuses on the problem of optimizing the trajectories of multiple searchers attempting to detect a non-evading moving target whose motion is conditionally deterministic. This problem is a parameter-distributed ...
