Larsen, Christopher J.
This project describes my work done in exploring different concepts of size for various infinite sets. The mathematical measurements discussed are cardinality, density, Lebesgue measure, Baire category and Hausdorff dimension. These principles are used to compare sizes of several infinite sets such as the set of real numbers, the set of rational numbers, the set of irrational numbers, the Cantor set, the generalized Cantor set, as well as sets described only by one of the earlier listed measurements of size, such as nowhere dense sets. Next, the consistency of these concepts is addressed. Examples are employed to show that a single set can be "large" under one consideration, but "small" under a second consideration. Also, mathematical proofs are used to show that a set may not take on certain combinations of size. For example, a set can not be nowhere dense and full measure. Lastly, sizes of some of the infinite sets are applied to problems in real analysis. It is proved that the upper Riemann integral of a positive function is positive, the Lebesgue integral of a positive, measurable function is positive, there is no function continuous exactly on the set of rational numbers, and the Hausdorff dimension of the Cantor set is In(2)/In(3).
Worcester Polytechnic Institute
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