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A sheaf theoretic approach to measure theory

Jackson, Matthew Tobias (2006) A sheaf theoretic approach to measure theory. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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 almost-everywhere equivalence is a Lawvere-Tierney 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 non-Boolean settings.Given a measure $mu$ on $mathcal{F}$, the Lawvere-Tierney 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 Radon-Nikodym theorem holds, so that for any locally finite $ullmu$, the Radon-Nikodym derivative $frac{du}{dmu}$ exists.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Jackson, Matthew Tobiasmajst46@pitt.eduMAJST46
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee CoChairHeath, Bobrwheath@pitt.eduRWHEATH
Committee MemberLennard, Chrislennard@pitt.eduLENNARD
Committee MemberScott,
Committee MemberGartside, Paulgartside@pitt.eduGARTSIDE
Committee MemberAwodey,
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:, etd-04202006-065320
Date Deposited: 10 Nov 2011 19:39
Last Modified: 15 Nov 2016 13:41


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