# A sheaf theoretic approach to measure theory

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

Keywords:mathematics

ISBN:

Date of Publication:06/02/2006