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1. Cohen, Michael Patrick. Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups.

Degree: 2013, University of North Texas

In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a countable union of Haar null sets in a Polish group is Haar null. We show the openly Haar null ideal may be distinct from the Haar null ideal, which resolves an uncertainty of Solecki. We show that compact sets are always Haar null in S? and in any countable product of locally compact non-compact groups, which extends the domain of a result of Dougherty. We show that any countable product of locally compact non-compact groups decomposes into the disjoint union of a meager set and a Haar null set, which gives a partial positive answer to a question of Darji. We display a translation property in the homeomorphism group Homeo+[0,1] which is impossible in any non-trivial locally compact group. Other related results are peppered throughout. Advisors/Committee Members: Kallman, Robert R., Gao, Su, Jackson, Steve, 1957-.

Subjects/Keywords: Topological groups; topological dynamics; Borel complexity; Haar null sets

…topology of pointwise convergence. Chapter 5 deals primarily with the σ-ideal of Haar null sets a… …openly Haar null sets. The theorems in these latter three chapters are new unless otherwise… …Theorem. 1.3. Haar Null Sets and Openly Haar Null Sets In Chapter 5 we investigate the general… …theory of Haar null sets as defined by Christensen, as well as the newer notion of openly Haar… …null sets as defined briefly by Solecki. It is a famous result of Haar/Weil/Cartan that every… 

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APA (6th Edition):

Cohen, M. P. (2013). Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc271792/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Cohen, Michael Patrick. “Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups.” 2013. Thesis, University of North Texas. Accessed August 13, 2020. https://digital.library.unt.edu/ark:/67531/metadc271792/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Cohen, Michael Patrick. “Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups.” 2013. Web. 13 Aug 2020.

Vancouver:

Cohen MP. Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups. [Internet] [Thesis]. University of North Texas; 2013. [cited 2020 Aug 13]. Available from: https://digital.library.unt.edu/ark:/67531/metadc271792/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Cohen MP. Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups. [Thesis]. University of North Texas; 2013. Available from: https://digital.library.unt.edu/ark:/67531/metadc271792/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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