Measure Theory

1 Overview

Measure theory makes precise the notion of size (length, area, mass, probability) for sets, and builds the integral on top of it. It underpins Probability Theory, where a probability measure is a measure of total mass one. This note is an index of the topics.

2 Fundamentals

• completion.
  • Generated Sigma Algebra
  • Product σ-Algebra
• Constructing measures
  • Outer Measure
  • Carathéodory Extension Theorem
  • Set Functions
• Generating tools for σ-algebras
  • Pi System
  • Dynkin's π-λ Theorem
  • Monotone Class Theorem

3 Lebesgue integration

• Measurable Function
• • Monotone Convergence Theorem
  • Fatou Lemma
  • Dominated Convergence Theorem
• Radon-Nikodym Derivative
• Limit Supremum & Limit Infimum

Backlinks

• Expectation
• Images and Preimages of Functions
• Mathematics
• Measurable Function
• Outer Measure
• Probability Spaces
• Vitalli Non-Measurability Theorem