Date of Graduation

Spring 2019

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Xingping Sun

Keywords

Hausdorff measure, Lebesgue measure, Cantor set, outer measure, Lebesgue-Stieltjes, measurable sets, Dirichlet function

Subject Categories

Analysis | Mathematics

Abstract

Measure theory is fundamental in the study of real analysis and serves as the basis for more robust integration methods than the classical Riemann integrals. Measure theory allows us to give precise meanings to lengths, areas, and volumes which are some of the most important mathematical measurements of the natural world. This thesis is devoted to discussing some of the major proofs and ideas of measure theory. We begin with a study of Lebesgue outer measure and Lebesgue measurable sets. After a brief discussion of non-measurable sets, we define Lebesgue measurable functions and the Lebesgue integral. In the last chapter we discuss general outer measures and give two specific examples of measures based on an outer measure and Carath\'eodory's definition of measurable sets. Lebesgue-Stieltjes measures are important because they are not limited to the identity function that gives rise to Lebesgue measure, and Hausdorff measure is often able to distinguish a variety of sets whose Lebesgue measure are all zero. The goal of this thesis is to present the proofs and ideas of fundamental measure theory in a way that is accessible and helpful to senior level undergraduates and beginning graduates in their quest to lay a solid foundation for further training in analysis.

Copyright

© Jacob N. Oliver

Open Access

Included in

Analysis Commons

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