Date of Submission

2-22-1986

Date of Award

2-22-1987

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Quantitative Economics

Department

Economic Research Unit (ERU-Kolkata)

Supervisor

Maitra, Ashok (ERU-Kolkata; ISI)

Abstract (Summary of the Work)

The present thesis consists of three parts.Part I of the thesis is devoted to the study of regularity propertion of classes of sets obtained from 8-8 operations applied to the class of clopen set s. In particular, the classes of set s obtained by applying the hierarchy of increasingly complex R-operations {Rp,), are investing at ed. Our points of departure in this study are the recent works on R- sets by Hi nm an and Burgess. In this thesis we have unified the methods of Hinman and Burgess by extensively u sing the game quantifier and methods of inductive definability. Using these tools we are able not only to considerably simplify the proofs of known results in the theory (e.g. the comparison of indices theorem on which so much of Lyapunov's work depend s as well as the category and 'measure formulas for R-set s) but also to obtain a number of new results for R- sets. Among these new re sulte we mention the measurable selection property for Ä‘ifferent levels of the hierarehy of R- set s, the scale property at all levels of the R-hierarchy (this was kn own only for the first two 1evels) and the uniformization prop erty at all levels of the R-hierarchy (again this was known only for the first level). These last mentioned results depend on the notion of presentability introduced by Burgess in his penetrating study of the measurable selection property of various families of set s. Our proof turns on the observation that the present- ability property holds at every level of the R-hierarchy. Finally, by combining analysic of the associated games with an adaptation of the Kechris Game Formula and the existence of measurable winning strategies; we are able to extend a recent result of Srivatsa on C-sets to R-sets. Briefly stated this result ensures that an R- set in the product of two Polish spaces can be approximated in the sense of category by means of sets in appropriate product o-field s, the importance of this result resting on the fact that setsin product o-field have a much more tractable structure than R- sets. Following Srivatsa we use this result to obtain a selection theorem of Burgess for R- sets.While Part I is devoted to the study of alassical (bold-f ace) R-set p, in Part II we take up the study of the effective analogues of the se classes at the finite levels. Using again the game theoretic methode of Moschovakie, we are able to establish various regularity properties of these offective classes. For instance, we show that aach o Rn. is a spector pointless. We also ob serve that various other results like the MeaBure and Category Formulas, the saale property hold in the effective case also. We also take another look at the problem of effectivizing C-sets. Hinman, by gener alizing Addison's construction of the effective Borel hierarchy, obtained a hierarchy of effective C-sets on w which exhaust 1sc ( E 1), the class of subsete ofrecursive in We, however, obtain an effective hierarchy in 1 sc (E,) in a different manner. We first exhibit a set D (T,a,x), where T varies over wellfounded w, which is universal for c-ante . When trees on * w, by taking sections of by wellfounded trees and reals of inoreasing complexity, we obtain a hierarchy whose scope is 180 (E1 ).

Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842862

Control Number

ISILib-TH296

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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