Date of Submission

9-28-1969

Date of Award

9-28-1970

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Supervisor

Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

It ts the nurposo of this chapter to collect together the problons that were acattored throughout the provious two chapters. This chaptor has sovon scctions. The first threo soc tions doal with the probloms that arc raised in or have c onnec tion witl: the topic of chaptor 1. The last tihree sectiona dcal with thc ne of chapter 2. Soction 4 stands mid- vay betweon bo tli, We ahall nunber the problomsi sorially and the nunber appears in bracko to. Wherover posaible wo shall includo somo dincuscion abcut the problcn involvod. There are no now rocults in thin chapter cyccpt perhapo one or two in the first section. Somotinos in what followo, wo shall if neccssary - invoko the axion of choice without explicit nention. Phrasca like one dɔes not knowIt is not knowm do of course nean that 'the prosent author does not know. It is alwoys of intercet - and nany times usoful - to know as to what propertics of Le be sguo ncaourable sets and functions have thoir counterparts in cete and functions with the ledre proporty. To clnborate thin point further, let denc to tiw colloucion of the Lobo rcue miasurablc sots on tho Toal line R and g the collootion of nota vith the Baire property. At tiuo ontset boti are traneintien inveri ont non- C ountably generated omal/sobran on R olored undor the Bouslin operation (See Kuratowalti (e]). Tho aounterparta of seta of 10 nauro Eoro in I nre tho se of tlhe firat catogory in g. This anal ogy go0e furthor. Bvery Lebesgue neaBurabio functi on (L.M.f) in Borol nonnuable outaide a null sot (All funotifona, unlcaa otlorwino stated to the contrary, trke thoir valuoe in thhe real lino oqui pped wi ti te umaal topology nd Borol field). Sinilariy sny function with t'io Briro property (B.P.f) is Borcl outeide a et of tlhe firat entogory. In fact moro precino versions are availablo - evory B.P.f ia continuous outsido a a ct of the firat category and any L.ll.fie of Baire class 1 outnidon null set. This đifforenee, howovor, aan bo explninod fron tho fact tlnt overy pet inio a Gamodulo a null wet whereas every net in in an opon sct nodulo a sot of the firat catogory. Thoorene 1, s, and 2, .of cliaptor 1 point nut that thio ne analogier go still deeper. Section 40 of Kuratowski (2) ahows tiovon in the occurrenco of tho patliologies oro in a sinilari ty. Wo nirll point sut two mare sinilgritien intor in thia noction - wiich though not ovrprisin, are bolieved to be now. It wouid be intoresting to find ctre analogion cndecoc the usce thoy can bo put to (P1 ).It is a striking but unfortunatcly not so well known the orem of Sicrpinsti[1) that thcre oxints a null sot A, in L such tint tho sun.

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:28843436

Control Number

ISILib-TH79

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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