Kan extensions in probability theory

Abstract

n this thesis we will discuss results and ideas in probability theory from a categorical point of view. One categorical concept in particular will be of interest to us, namely that of Kan extensions. We will use Kan extensions of ‘ordinary’ functors, enriched functors and lax natural transformations to give categorical proofs of some fundamental results in probability theory and measure theory. We use Kan extensions of ‘ordinary’ functors to represent probability monads as codensity monads. We consider a functor representing probability measures on countable spaces. By Kan extending this functor along itself, we obtain a codensity monad describing probability measures on all spaces. In this way we represent probability monads such as the Giry monad, the Radon monad and the Kantorovich monad.

Kan extensions of lax natural transformations are used to obtain a categorical proof of the Carath´eodorody extensions theorem. The Carath´eodory extension theorem is a fundamental theorem in measure theory that says that premeasures can be extended to measures. We first develop a framework for Kan extensions of lax natural transformations. We then represent outer and inner (pre)measures by certain lax and colax natural transformations. By applying the results on extensions of transformations a categorical proof of Carath´eodory’s extension theorem is obtained.

We also give a categorical view on the Radon–Nikodym theorem and martingales. For this we need Kan extensions of enriched functors. We start by observing that the finite version of the Radon–Nikodym theorem is trivial and that it can be interpreted as a natural isomorphism between certain functors, enriched over CMet, the category of complete metric spaces and 1-Lipschitz maps. We proceed by Kan extending these, to obtain the general version of the Radon–Nikodym theorem. Concepts such as conditional expectation and martingales naturally appear in this construction. By proving that these extended functors preserve certain cofiltered limits, we obtain categorical proofs of a weaker version of a martingale convergence theorem and the Kolmogorov extension theorem.