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On Monadic Vector-Valued Integration

Benedikt Peterseim

Abstract

In recent times, there has been a growing interest in a structuralist understanding of probability, measure and integration theory. The present thesis contributes to this programme in three ways. First, we construct a commutative probability monad on the cartesian closed category of hk-spaces (also known as CGWH spaces, or weak Hausdorff k-spaces in the literature). Secondly, in order to achieve this in a seamless way, we develop the theory of paired linear hk-spaces, a functional-analytic category tailored to the duality between measures and functionals. Finally, vector-valued integration emerges naturally from the free-forgetful adjunction between paired linear hk-spaces and hk-spaces, inducing a commutative monad of compactly supported measures and leading to a theory of monadic vector-valued integration.

On Monadic Vector-Valued Integration

Abstract

In recent times, there has been a growing interest in a structuralist understanding of probability, measure and integration theory. The present thesis contributes to this programme in three ways. First, we construct a commutative probability monad on the cartesian closed category of hk-spaces (also known as CGWH spaces, or weak Hausdorff k-spaces in the literature). Secondly, in order to achieve this in a seamless way, we develop the theory of paired linear hk-spaces, a functional-analytic category tailored to the duality between measures and functionals. Finally, vector-valued integration emerges naturally from the free-forgetful adjunction between paired linear hk-spaces and hk-spaces, inducing a commutative monad of compactly supported measures and leading to a theory of monadic vector-valued integration.
Paper Structure (157 sections, 139 theorems, 288 equations, 4 tables)

This paper contains 157 sections, 139 theorems, 288 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Conventions
  3. Main Results
  4. Background and Motivation
  5. The analysis of "functionals" and cartesian closed categories of topological spaces
  6. Probability and measure monads
  7. From Polish spaces to cartesian closed categories of spaces
  8. Some foundational problems of the notion of topological vector space
  9. Unavoidable Discontinuity
  10. Tensor troubles
  11. Overview
  12. Linear $hk$-spaces and linear QCB spaces
  13. Replete linear $hk$-spaces
  14. Smith duality
  15. Paired linear $hk$-spaces
  16. ...and 142 more sections

Key Result

Theorem 1.2.0.1

The category $\mathsf{L}$ of paired linear $hk$-spaces admits a closed symmetric monoidal structure consisting of a tensor product $\,\widehat{\otimes}\,$ and an internal hom $[-,-]$. For paired linear $hk$-spaces $V,W$, the internal hom $[V,W]$ is the closed subspace of $C(V,W)$ consisting of morph a natural isomorphism of paired linear $hk$-spaces. Moreover, for every paired linear $hk$-space $V

Theorems & Definitions (390)

  • Theorem 1.2.0.1
  • Theorem 1.2.0.2
  • Theorem 1.2.0.3
  • Definition 1.3.1.1
  • Definition 1.3.1.2: $k$-space, $hk$-space
  • Definition 1.3.1.4
  • Definition 1.3.2.1: Monad
  • Example 1.3.2.2: The Giry monad on Polish spaces, giry1981categorical
  • Definition 1.4.1.1
  • Remark 1.4.1.2
  • ...and 380 more