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(Unpublished)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 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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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| 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/etd-04202006-065320/, etd-04202006-065320 |
| Date Deposited: |
10 Nov 2011 19:39 |
| Last Modified: |
15 Nov 2016 13:41 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/7348 |
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