Citation
Kirk, Ronald Brian (1968) Measures in topological spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/00H36K91. https://resolver.caltech.edu/CaltechETD:etd09252002093739
Abstract
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Let X be a completelyregular topological space and let C*(X) denote the space of all bounded, realvalued continuous functions on X. For a positive linear functional [...] on C*(X), consider the following two continuity conditions. [...] is said to be a Bintegral if whenever [...] and [...] for all [...], then [...]. [...] is said to be Bnormal if whenever [...] is a directed system with [...] for all [...], then [...]. It is obvious that a Bnormal functional is always a Bintegral. The main concern of this paper is what can be said in the converse direction.
Methods are developed for discussing this question. Of particular importance is the representation of C*(X) as a space [...] of finitelyadditive set functions on a certain algebra of subsets of X. This result was first announced by A. D. Alexandrov, but his proof was obscure. Since there seem to be no proofs readily available in the literature, a complete proof is given here. Supports of functionals are discussed and a relatively simple proof is given of the fact that every Bintegral is Bnormal if and only if every Bintegral has a support.
The space X is said to be Bcompact if every Bintegral is Bnormal. It is shown that Bcompactness is a topological invariant and various topological properties of Bcompact spaces are investigated. For instance, it is shown that Bcompactness is permanent on the closed sets and the cozero sets of a Bcompact space. In the case that the spaces involved are locallycompact, it is shown that countable products and finite intersections of Bcompact spaces are Bcompact.
Also Bcompactness is studied with reference to the classical compactness conditions. For instance, it is shown that if X is Bcompact, then X is realcompact. Or that if X is paracompact and if the continuum hypothesis holds, then X is Bcompact if and only if X is realcompact.
Finally, the methods and results developed in the paper are applied to formulate and prove a very general version of the classical Kolmogorov consistency theorem of probability theory. The result is as follows. If X is a locallycompact, Bcompact space and if S is an abstract set, then a necessary and sufficient condition that a finitelyadditive set function defined on the Baire (or the Borel) cylinder sets of X[superscript S] be a measure is that its projection on each of the finite coordinate spaces be Baire (or regular Borel) measures.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  8 April 1968 
Record Number:  CaltechETD:etd09252002093739 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd09252002093739 
DOI:  10.7907/00H36K91 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3748 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  26 Sep 2002 
Last Modified:  21 Dec 2019 04:44 
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