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Articles 1 - 5 of 5
Full-Text Articles in Physical Sciences and Mathematics
The Banach-Tarski Paradox, Matthew Jacob Norman
The Banach-Tarski Paradox, Matthew Jacob Norman
Theses Digitization Project
The purpose of this thesis is to establish the history and motivation leading up to the Banach-Tarski Paradox, as well as its proof. This study discusses the early history of set theory as it is documented as well as the necessary basics of set theory in order to further understand the contents within. Set theory not only proved to be for the mathematical at heart but also struck interest into the mind of philosophers, theologians, and logicians.
A Comparison Of Category And Lebesgue Measure, Adam Matthew Moore
A Comparison Of Category And Lebesgue Measure, Adam Matthew Moore
Theses Digitization Project
This study, Lebesgue measure and category have proved to be useful tools in describing the size of sets. The notions of category and Lebesgue measure are commonly used to describe the size of a set of real numbers (or of a subset of Rn). Although cardinality is also a measure of the size of a set, category and measure are often the more important gauges of size when studying properties of classes of real functions, such as the space of continuous functions or the space of derivatives.
The Riesz Representation Theorem For Linear Functionals, Thomas Daniel Schellhous
The Riesz Representation Theorem For Linear Functionals, Thomas Daniel Schellhous
Theses Digitization Project
This study will investigate the Riesz representation theorem for linear functionals in relation to locally compact Hausdorff spaces. Two other theorems that are commonly called "Riesz representation theorem" are the theorem for finite-dimensional inner product spaces and the theorem for Hilbert spaces [BN00], and studying these interesting topics helps us to not only gain a better understanding of how linear functionals interact with vector spaces over which they are defined, but also to see faint threads that hint at a deep connection between the various fields of modern mathematics.
Hausdorff Dimension, Loren Beth Nemeth
Hausdorff Dimension, Loren Beth Nemeth
Theses Digitization Project
The purpose of this study was to define topological dimension and Hausdorff dimension, Namely metric space theory and measure theory. It was verified that in the sets of elementary geometry, the dimensions agree, while in the case of the fractals, the Hausdorff dimension is strictly larger than the topological dimension.
From Measure To Integration, Sara Hernandez Mcloughlin
From Measure To Integration, Sara Hernandez Mcloughlin
Theses Digitization Project
The thesis studies the notions of outer measure, Lebesgue measurable sets and Lebesgue measure, in detail. After developing Lebesgue integration over the real line, the Riemann integrable functions are classified as those functions whose set of points of discontinuity has measure zero. The convergence theorems are proven and it is shown how these theorems are valid under less stringent assumptions that are required for the Riemann integral. A detailed analysis of abstract measure theory for general measure spaces is given.