Ameen, Zanyar
(2015)
*FINITELY ADDITIVE MEASURES ON
TOPOLOGICAL SPACES AND BOOLEAN
ALGEBRAS.*
Doctoral thesis, University of East Anglia.

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## Abstract

The thesis studies some problems in measure theory. In particular, a possible generalization

corresponding to Maharam Theorem for �nitely additive measures (charges).

In the �rst Chapter, we give some de�nitions and results on di�erent areas of Mathematics

that will be used during this work.

In Chapter two, we recall the de�nitions of nonatomic, continuous and Darboux

charges, and show their relations tWe hereby con�rm that all work without reference are

our work excluding the chapter one. We hereby con�rm that all work without reference

are our work excluding the chapter one. o each other. The relation between charges

on Boolean algebras and the induced measures on their Stone spaces is mentioned

in this chapter. We also show that for any charge algebra, there exists a compact

zero-dimensional space such that its charge algebra is isomorphic to the given charge

algebra.

In Chapter three, we give the de�nition of Jordan measure and some of its outcomes.

We de�ne another measure on an algebra of subsets of some set called Jordanian measure,

and investigate it. Then we de�ne the Jordan algebras and Jordanian algebras,

and study some of their properties.

Chapter four is mostly devoted to the investigation of uniformly regular measures

and charges (on both Boolean algebras and topological spaces). We show how the

properties of a charge on a Boolean algebra can be transferred to the induced measure

on its Stone space. We give a di�erent proof to a result by Mercourakis in [36, Remark

1.10]. In 2013, Borodulin-Nadzieja and Dºamonja [10, Theorem 4.1] proved the countable

version of Maharam Theorem for charges using uniform regularity. We show that

3

4

this result can be proved under weaker assumption and further extended.

The �nal Chapter is concerned with the higher versions of uniform regularity which

are called uniform �-regularity. We study these types of measures and obtain several

results and characterizations. The major contribution to this work is that we show

we cannot hope for a higher analogue of Maharam Theorem for charges using uniform

�-regularity. In particular, Theorem 4.1 in [10] and Remark 1.10 in [36] cannot be

extended for all cardinals. We prove that a higher version of Theorem 4.1 in [10]

(resp. Remark 1.10 in [36]) can be proved only for charges on free algebras on � many

generators (resp. measures on a product of compact metric spaces). We also generalize

Proposition 2.10 in [26].

Item Type: | Thesis (Doctoral) |
---|---|

Faculty \ School: | Faculty of Science > School of Mathematics |

Depositing User: | Users 2259 not found. |

Date Deposited: | 29 Jan 2016 09:33 |

Last Modified: | 29 Jan 2016 09:33 |

URI: | https://ueaeprints.uea.ac.uk/id/eprint/56864 |

DOI: |

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