#### Date of Graduation

Summer 2011

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Xingping Sun

#### Keywords

gauge, integration, Henstock, Kurzweil, Lebesgue integral

#### Subject Categories

Mathematics

#### Abstract

The goal of this thesis is to exposit a simpler method of generalizing the concept of Riemann integration than the standard-bearer way. The generalization used is the gauge integral, also known as the Henstock-Kurzweil integral. This integral has a definition that is a simple generalization of the Riemann integral and is shown to satisfy the Monotone and Dominated Convergence Theorems. This thesis is an amalgamation of work by different authors, made complete by including proofs for every result. A new definition for a gauge of the real number line is included which leads to a more intuitive approach to integration over unbounded intervals. We give new and simpler proofs of the general convergence theorems that require a minimal amount of measure theory. Interesting, well-known results proved in this thesis are the Fundamental Theorem of Calculus in full generality, the convergence theorems, and the equivalence of Lebesgue integrability to absolute gauge integrability.

#### Copyright

© Andrew Thomas Swift

#### Recommended Citation

Swift, Andrew Thomas, "The Gauge Integral and Its Connection to the Lebesgue Integral" (2011). *MSU Graduate Theses*. 1639.

https://bearworks.missouristate.edu/theses/1639