Jackson, Matthew Tobias
(2006)
A sheaf theoretic approach to measure theory.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
The topos $extrm{Sh}(mathcal{F})$ of sheaves on a $sigma$algebra $mathcal{F}$ is a natural home for measure theory. The collection of measures is a sheaf, the collection of measurable real valued functionsis a sheaf, the operation of integrationis a natural transformation, and the concept of almosteverywhere equivalence is a LawvereTierney topology. The sheaf of measurable real valued functions is the Dedekind real numbers object in $extrm{Sh}(mathcal{F})$ and the topology of ``almost everywhere equivalence`` is the closed topologyinduced by the sieve of negligible sets The other elements of measure theory have not previously been described using the internallanguage of $extrm{Sh}(mathcal{F})$. The sheaf of measures, and the natural transformation of integration, are here described using the internal languages of $extrm{Sh}(mathcal{F})$ and $widehat{mathcal{F}}$, the topos of presheaves on $mathcal{F}$.These internal constructions describe corresponding components in any topos $mmathscr{E}$ with a designatedtopology $j$. In the case where $mmathscr{E}=widehat{mathcal{L}}$ is the topos of presheaves on a locale, and$j$ is the canonical topology, then the presheaf of measures is a sheaf on $mathcal{L}$. A definition of the measure theory on $mathcal{L}$ is given, and it is shown that when$extrm{Sh}(mathcal{F})simeqextrm{Sh}(mathcal{L})$, or equivalently, when $mathcal{L}$ is the locale of closed sieves in $mathcal{F}$this measure theory coincides with the traditional measure theory of a $sigma$algebra $mathcal{F}$.In doing this, the interpretation of the topology of ``almost everywhere' equivalence is modified so as to better reflect nonBoolean settings.Given a measure $mu$ on $mathcal{F}$, the LawvereTierney topology that expressesthe notion of ``$mu$almost everywhere equivalence' induces a subtopos $extrm{Sh}_{mu}(mathcal{L})$. If this subtopos is Boolean, and if $mu$ is locally finite, then the RadonNikodym theorem holds, so that for any locally finite $ullmu$, the RadonNikodym derivative $frac{du}{dmu}$ exists.
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Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
2 June 2006 
Date Type: 
Completion 
Defense Date: 
13 April 2006 
Approval Date: 
2 June 2006 
Submission Date: 
20 April 2006 
Access Restriction: 
No restriction; Release the ETD for access worldwide immediately. 
Institution: 
University of Pittsburgh 
Schools and Programs: 
Dietrich School of Arts and Sciences > Mathematics 
Degree: 
PhD  Doctor of Philosophy 
Thesis Type: 
Doctoral Dissertation 
Refereed: 
Yes 
Uncontrolled Keywords: 
logic; sheaf theory; topos theory 
Other ID: 
http://etd.library.pitt.edu/ETD/available/etd04202006065320/, etd04202006065320 
Date Deposited: 
10 Nov 2011 19:39 
Last Modified: 
15 Nov 2016 13:41 
URI: 
http://dscholarship.pitt.edu/id/eprint/7348 
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