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Lecture notes: Probability with Measure

Matija Vidmar

Abstract

Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of Lebesgue integration are developed: convergence theorems, product spaces and Tonelli-Fubini, indefinite integration and absolute continuity, L-spaces and integral inequalities. Everything is set up so that in the second part the fundamental concepts of probability (such as those of random elements and their laws, independence, conditioning) can be cast swiftly in the measure-theoretic setting. Some emphasis is placed on monotone class and Dynkin's lemma type arguments. Products of arbitrary families of probabilities and Kolmogorov's extension theorem are treated.

Lecture notes: Probability with Measure

Abstract

Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of Lebesgue integration are developed: convergence theorems, product spaces and Tonelli-Fubini, indefinite integration and absolute continuity, L-spaces and integral inequalities. Everything is set up so that in the second part the fundamental concepts of probability (such as those of random elements and their laws, independence, conditioning) can be cast swiftly in the measure-theoretic setting. Some emphasis is placed on monotone class and Dynkin's lemma type arguments. Products of arbitrary families of probabilities and Kolmogorov's extension theorem are treated.

Paper Structure

This paper contains 22 sections, 61 theorems, 70 equations.

Table of Contents

  1. Measure
  2. Measurability and measures
  3. Measurable sets
  4. Measures
  5. Measurable maps and generated $\sigma$-fields
  6. Borel sets of the extended real line and Borel measurability of numerical functions
  7. Monotone class arguments
  8. Lebesgue-Stieltjes measures
  9. Completeness of measures
  10. Outer measures and Carathéodory's extension theorem
  11. Integration on measurable spaces
  12. The Lebesgue integral
  13. Convergence theorems with some immediate corollaries
  14. Results concerning the interchange of the order of integration
  15. Indefinite integration and absolute continuity
  16. ...and 7 more sections

Key Result

Proposition 1.6

Let $\mathcal{A}\subset 2^\Omega$ be closed for $\mathsf{c}^\Omega$ and let $\emptyset\in \mathcal{A}$. Then $\mathcal{A}$ is a $\sigma$-algebra on $\Omega$ iff $\mathcal{A}$ is closed for $\sigma\cap$, in which case $\Omega\in \mathcal{A}$ and $\mathcal{A}$ is closed for $\cap$, $\cup$ and $\backsl

Theorems & Definitions (255)

  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Remark
  • Proposition 1.6
  • proof
  • Definition 1.7
  • Example 1.8
  • ...and 245 more