#### Date of Submission

2-28-1972

#### Date of Award

2-28-1973

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Economic Research Unit (ERU-Kolkata)

#### Supervisor

Maitra, Ashok (ERU-Kolkata; ISI)

#### Abstract (Summary of the Work)

This Thesis is divided into six chapters the first three chapters constituting studies in Boolean algebras and the last three chapters constituting studies in measure theory. We give below a sketch of the main problems treated in this Thesis.A De composition theorem due to Sobczyk and Hammer (28] implies that strongly continuous charges on Boolean algebras play a role similar to that of nonatomic measures on Boolean g-algebras. CHAPTER 1. Rudin [25] and Knowles [14] gave necessary and sufficient conditions for the Boral o-ficld of a compact Hausdorff space to admit a nonatomic measure. But there are no necessary and sufficient algebraic conditions for a Boolean o-algebra to admit a nonatomic In this chapter we solve the analogous problem of exis- measure, tence of strongly continuous charges on Boolean algebras. examine the richnoss of strongly continuous charges in the space We also of all charges on a Boolcan algebra.CHAPTER 2. Given any charge on a Boolcan algebra, equivalently, given any charge spaco (Î©, A, H) one associates a natural metric space (A(u), du) with it. A naturel question that arises is: How far the topological propertics of the metric space (A(H), d) reflect on A and u? This problem is treated in this chapter. A satisfactory picture omerges when (Î©, A, H) is a moasure space. When (Î©. 4, H) is a charge space the problem is solved Partially.CHAPTER 3: The contents of this chapter are inspirod by a papor of B, V, Rao [23]. Fcr any Boolenn algebra A the class of all subalgobras of forms a compicte lattice which we call L.. The question of distributivity of L, is not vcry interesting becausc it hes n trivial solution. In this chn pter we deal with the complementation in L. i characterisation of L, is a complemented lattice is still lacking. In addition to the study of complomentation in L A such that we also gonaralize B.v.Raols results in scvcral dircetions.CHAPTER 4: The origin of Chapter 4 is a paper of v. Ficker [7] - N whore (Î©, B, H) is a measure space and N 1s the colloc- who attomptcd to charactcrisc countable chain condition in tion of all H-null scts. We damonstrato that the main the orom of [7] is incorrect and prove a strongor version of Ficker's theorem for certain typos of measures.CHPTER 5: The problem of this chapter was suggested by Profcssor M. G. Nadkarni. The problem is : Gi en two ral valucd measurable functions f and dofined on a Borol structurG (x, B) when does there exist a nonatomic probability measure g independent. red the problem has a trivial solution. Here we solve this which makes f and If nonatomicity is not requi- problem when X is the real line and B is the Borel o-algebra of the real line. We consider some extensions also

#### Control Number

ISILib-TH14

#### Creative Commons 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

#### Recommended Citation

Rao, K. P. S. Bhaskara Dr., "Studies in Boolean Algebras and Measure Theory." (1973). *Doctoral Theses*. 114.

https://digitalcommons.isical.ac.in/doctoral-theses/114

## 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:28842890