Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces

Abstract

Bishop style constructive integration theory constitutes an important milestone in constructive mathematics as it demonstrates the actual feasibility of developing a rich theory of integration within the constructive framework. However, Bishop’s approach has the fundamental flaw that it allows impredicativity in the sense that it uses statements that contain quantification over the whole universe of sets. In this thesis, I work towards amending Bishop’s theory in order to remove this impredicativity. Furthermore, I try to increase clarity through the explicit use of moduli. First, I introduce the necessary fundamental notions of Bishop Set Theory as presented by Petrakis. Following Bishop’s book, I develop the theory of locally compact metric spaces. Lastly, I introduce a notion of integration on locally compact metric spaces and prove that the set of partial functions with compact support constitute an integration space in a sensible manner.

Cite

Hayagriva YAML

cp-integration-lcms:
  type: Thesis
  title: Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces
  author: Grubmüller, Fabian Lukas
  date: 2022-06-06
  organization: Ludwig-Maximilians-Universität München
  url: "https://flgrubm.github.io/theses/2022-06-06-bt-cp-integration"
  note: Bachelor thesis

BibTeX

@mastersthesis{cp-integration-lcms,
    type = {Bachelor thesis},
    title = {Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces},
    author = {Grubm\"{u}ller, Fabian Lukas},
    year = {2022},
    school = {LMU M\"{u}nchen},
    note = {\url{https://flgrubm.github.io/theses/2022-06-06-bt-cp-integration}},
}

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Notes

Type: Bachelor Thesis
Supervisor: Dr. Iosif Petrakis
Programme: Mathematics B.Sc.
University: Ludwig-Maximilians-Universität München