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Some selector theorems for nonseparable metric spaces
Thesis (M. Sc.) -- University of Stellenbosch, 1990.
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| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2012
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| _version_ | 1867614015908216834 |
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| access_status_str | Open Access |
| author | Brink, Harry Edward |
| author2 | Maritz, P. |
| author_browse | Brink, Harry Edward Maritz, P. |
| author_facet | Maritz, P. Brink, Harry Edward |
| author_sort | Brink, Harry Edward |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (M. Sc.) -- University of Stellenbosch, 1990. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/67359 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:45:19.124Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/67359 Some selector theorems for nonseparable metric spaces Brink, Harry Edward Maritz, P. Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Metric spaces Dissertations -- Mathematics Thesis (M. Sc.) -- University of Stellenbosch, 1990. The classical measurable selector theorem of Kuratowski and Ryll-Nardzewski states that, if F: T → X is a lower measurable multifunction, defined on a measurable space T and taking non-empty closed values in a complete separable metric space X, then F has a measurable selector. In order to generalize the above mentioned theorem to the case when X is a nonseparable metric space, additional assumptions are necessary. Some selector theorems for nonseparable metric spaces are known, however some form of separability is retained, for example by imposing certain smallness conditions on the values of the multifunction, such as compactness and separability. If the measurable space T is reasonably nice, such as in the case of a metric space T with its a-algebra of Borel sets, useful nonseparable extensions of the above fundamental selector theorem are known for compact-valued multifunctions (see [8]). For separable-valued multifunctions, similar results are known, but only in special cases or under additional set theoretic axioms (see [9] and [22]). One of the main topological problems on selectors is to determine whether there exists a selector having some "nice" topological properties. The existence of measurable and Borel-selectors are very important and has applications in mathematical economics, control theory, operator theory and statistics (see [23) and [17]). My intention with this present thesis is to give an outline of the work done on selector theorems in the case of nonseparable metric spaces. For this purpose we discuss the papers of Himmelberg [11], Kuratowski and Ryll-Nardzewski [ 16] and Himmelberg Van Vleck and Prikry [12]. Our approach is as follows : In Chapter I the definitions of upper and lower semi-continuous multifunctions F: T → X are given and their relationship with open and closed sets in X are investigated. The main theorem is Theorem 2.11 which is a combination of Propositions 2.9 and 2.10. These two propositions give us conditions on T and X under which F is of Baire class 1. In §3 it is shown that the Vietoris (exponential) topology is a topology for I, the class of all non-empty closed sets in a T1- topological space U. In addition, some relations between U and I and U and K, with K all non-empty compact sets in 1, are given. In the first half of Chapter II the measurability of multifunctions is developed and the logical relations among the various definitions of measurability are worked out . The relationship between Debrea's [4 ] criteria for the measurability of a multifunction and our criterion, namely Definition 1.2 is studied in §3 . The main theorem is Theorem 3.5 which follows directly from Proposition 3.4. In Chapter III the general theorem on selectors by Kuratowski and Ryll-Nardzewski is discussed and some modifications by Himmelberg (11] and Engelking (5] of the above mentioned selecter theorem are also stated. Measurable selectors for closed set valued maps into arbitrary complete metric spaces, independent of any separability assumptions, are constructed in the final chapter. As mentioned before, several authors have done work on removing the separability assumption (see [8] and [13]). The main difference between their and Himmelberg, Van Vleck and Prikry's approach is that the former tend to retain some form of separability (e.g. compact, hence separable, values in an arbitrary, not necessarily separable space) whereas the latter allow completely arbitrary closed sets as values. In [10] Hansell took the study of measurable selectors in nonseparable spaces further by adopting a method of approach different from that of Himmelberg, Van Vleck and Prikry. To explain a few notations, we have the following: if A and B are subsets of a given set, then set-theoretic inclusion and substraction will be denoted by Ac B and A\B respectively. The symbol .. is read as: "if and only if". We denote by iN the set of all natural numbers and we refer the reader to [6] for the definitions of concepts not explicitly given in the text. Masters 2012-08-27T12:09:48Z 2012-08-27T12:09:48Z 1990 Thesis http://hdl.handle.net/10019.1/67359 en_ZA Stellenbosch University 89 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Metric spaces Dissertations -- Mathematics Brink, Harry Edward Some selector theorems for nonseparable metric spaces |
| title | Some selector theorems for nonseparable metric spaces |
| title_full | Some selector theorems for nonseparable metric spaces |
| title_fullStr | Some selector theorems for nonseparable metric spaces |
| title_full_unstemmed | Some selector theorems for nonseparable metric spaces |
| title_short | Some selector theorems for nonseparable metric spaces |
| title_sort | some selector theorems for nonseparable metric spaces |
| topic | Metric spaces Dissertations -- Mathematics |
| url | http://hdl.handle.net/10019.1/67359 |
| work_keys_str_mv | AT brinkharryedward someselectortheoremsfornonseparablemetricspaces |