Abstract Measure Theory

In 1902 the French mathematician Henri Lebesgue introduced his revolutionary theory of integration, building on and resolving many of the issues of Riemann's integration theory while keeping much of its intuition and utility.
Beyond its immediate success, the structures and ideas Lebesgue introduced lent themselves to generalisation and birthed what we now know as (Abstract) Measure Theory .
Measure theory was instrumental in the development and advancement of many mathematical fields including Probability Theory, Partial Differential Equations, Functional Analysis, and Ergodic Theory.

The basic setting in Measure theory is a triplet \((X,\mathcal{F},\mu)\) where \(X\) is a set which represents the setting we're in, \(\mathcal{F}\) is a collection of subsets of \(X\) we can "size up" and satisfies

• \(\emptyset\in \mathcal{F}\),
• If \(E\in \mathcal{F}\) then \(E^c\in \mathcal{F}\),
• If \(\{E_n\}_{n\in\mathbb{N}}\subset \mathcal{F}\) then \(\cup_{n\in\mathbb{N}}E_n\in \mathcal{F}\),

• \(\mu(\emptyset)=0\),
• If \(\{E_n\}_{n\in\mathbb{N}}\subset \mathcal{F}\) are disjoint then $$\mu\left(\cup_{n\in\mathbb{N}}E_n \right) = \sum_{n\in\mathbb{N}}\mu(E_n).$$

Our goal in this project will be to explore abstract measure theory. We will cover topics such as

• The structure of measurable sets: Algebras and \(\sigma-\)Algebras.
• Outer measures and measures. Properties of measures and completion of measures.
• Measurable maps and integration.
• Product measures and the Fubini-Tonelli theorem.

• Modes of convergence of measures.
• Complex valued measures.
• The Radon-Nikodum Theorem.
• Riesz–Markov–Kakutani representation theorem.
• Kuratowski–Ryll-Nardzewski measurable selection theorem.
• Disintegration of measures.

Prerequisites : Analysis III (essential) and Complex Analysis II.

Useful Prerequisites (not mandatory but helpful): Topology II.

Potential co-requisite (not mandatory but helpful): Functional Analysis and Applications IV.

• B. Heinz: Measure and Integration Theory.
• W. Rudin: Real and Complex Analysis.
• G. B. Folland: Real Analysis - modern techniques and
• R. G. Bartle: The Elements of Integration and Lebesgue Measure.
• M.R. Leadbetter, S. Cambanis, V. Pipiras: A basic course in measure and probability - theory for applications.
• A. Klenke: Probability theory - a comprehensive course.