A sheaf theoretic approach to measure theory

by Jackson, Matthew Tobias

Abstract (Summary)
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 functions is a sheaf, the operation of integration is 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 topology induced by the sieve of negligible sets The other elements of measure theory have not previously been described using the internal language 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 designated topology $j$. 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})simeq extrm{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 expresses the 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 $frac{d u}{dmu}$ exists.
Bibliographical Information:

Advisor:Chris Lennard; Dana Scott; Paul Gartside; Bob Heath; Steve Awodey

School:University of Pittsburgh

School Location:USA - Pennsylvania

Source Type:Master's Thesis



Date of Publication:06/02/2006

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