Measure Theory

Here are links to my Measure Theory screencasts from 2011-12, and the associated PDF slides. This is Chapter 3 of my Level 4 module Further Topics in Analysis (G14FTA/MATH4047).

See also http://wp.me/posHB-da for links to a complete set of materials from a full module I gave on Measure and Integration in the years 2006-7-8.

As usual, I had a lot of “fun” with attempting to record additional webcam footage of myself. See some posts in my blog such as
https://explainingmaths.wordpress.com/2012/02/15/camtasia-problems/
for example. However, I was quite impressed by the webcam’s “follow-my-face” software, when it worked! (See Section 3.5 onwards below.)

Measure Theory

(Chapter 3 from G14FTA Further Topics in Analysis 2011-12)

Suitable for students with some knowledge of metric and topological spaces.

Brief description: In Measure Theory we look carefully at various ways to measure the size of a set. The theory makes rigorous the notions of length, area and volume, and generalises these notions. Measure Theory, along with the associated theory of (Lebesgue) integration, has important applications in many areas, including Functional Analysis, Harmonic Analysis and Probability Theory.

Slides (see also screencasts below)

Screencasts

These screencasts were recorded using Camtasia on my Toshiba tablet PC, with the assistance of a Logitech Webcam Pro 9000 for the audio and  webcam footage. I used Bluebeam PDF Revu to annotate the pre-prepared PDF skeleton slides.

Two of the screencast recordings originally failed (see below), but in 2017 I narrated additional screencasts to cover the material on those slides.

The original recordings require flash to play them. However I am now linking (below) to versions on the University of Nottingham’s MediaSpace server. I recently finished correcting the captions (subtitles) for these.
Thanks to Debs Storey for adding in current University branding to the MediaSpace versions of the videos! 

These videos are also available on YouTube as individual videos and as a Measure Theory Playlist.

  • Section 3.1, The Extended Real Line

    • Part 1 of 15: Slides 20-23
      An introduction to the extended real line, with its total order (extending the total order on the real line), and a metric (induced, via a suitable bijection, by the usual metric on [-1,1]).
      Brief discussion of convergence in the extended real line.
    • Part 2 of 15: Slides 23-24
      Discussion of convergence to infinity for sequences of non-negative extended real numbers.
      Monotone sequences (nondecreasing or nonincreasing) and the Monotone Sequence Theorem for the extended real line.

  • Section 3.2, Arithmetic and series in the non-negative extended real numbers

    • Part 3 of 15: Slides 25-28
      Addition and multiplication for non-negative extended real numbers. Problems with the cancellation laws. Series of non-negative extended real numbers.
      Some discussion of problems with the algebra of limits.

  • Section 3.3, The algebra of limits

    • Part 4 of 15: Slides 29-30
      The algebra of limits for sums of sequences of non-negative extended real numbers.
      The algebra of limits for products, and its limitations, for sequences of non-negative extended real numbers.

  • Section 3.4, Collections of sets

    • Part 5 of 15: Slides 31-35
      Sets of sets, which we call collections of sets.
      Comparison of the properties of some collections of sets, including topologies, and the collection of closed sets in a topological space.
      Definition,  properties and examples of sigma-algebras.
    • Part 6 of 15: Slides 35-36
      Audio narration, with digital pointer, to go with annotated slides, as the original recording failed
      More examples of σ-algebras: σ-algebras generated by finite collections of sets; sets which are countable or have countable complement; optional discussion of sets with the Property of Baire.

  • Section 3.5: The sigma-algebra generated by a collection of subsets

    • Part 7 of 15: Slide 37
      Audio narration, with digital pointer, to go with annotated slides, as the original recording failed
      Brief discussion of how to generate a topology from a collection of sets (as a sub-base), and how it is harder to give such a concrete approach to generating a σ-algebra. The intersection of any family of σ-algebras on X is still a σ-algebra on X. The definition of the σ-algebra on a set X generated by a collection of subsets of X. Brief discussion relating this to open sets and sets with the Property of Baire.
    • Part 8 of 15: Slides 37-40 (notable for the first successful use of my webcam’s follow-my-face software in a lecture!)
      Properties of the sigma-algebra on a set X generated by a collection of subsets.
      Methods for showing inclusion or equality of sigma-algebras generated by two different collections of subsets.
      The Borel sets in the real line: definition and examples.

  • Section 3.6: Measurable spaces, measurable sets, measures and measure spaces

    • Part 9 of 15: Slides 41-43
      Measurable spaces and measurable sets. Brief discussion of length, area and volume, the idea behind Lebesgue measure, and some of the issues.
      The definition of a (non-negative) measure on a sigma-algebra.
    • Part 10 of 15: Slides 41-47
      Terminology: countable additivity, pairwise disjoint unions. Notation for pairwise disjoint unions. Examples and constructions of measures: counting measure; combinations of measures; the biggest possible measure; point-mass measures.

  • Section 3.7: Properties of measures

    • Part 11 of 15: Slides 48-53
      Deduction of standard properties of measures from the two  axioms: finite additivity; monotonicity; countable subadditivity.
      Continuity properties of measures stated and discussed.
    • Part 12 of 15: Slides 52-53
      Proofs of the continuity properties of measures for nested increasing unions and nested decreasing intersections of sequences of measurable sets.

  • Section 3.8: Lebesgue outer measure and Lebesgue measure on the real line

    • Part 13 of 15: Slides 54-56
      Brief discussion of the following: what we hope to achieve when measuring length, area and volume; the Banach-Tarski paradox; the Carathéodory extension theorem.
      The definition of Lebesgue outer measure λ* on the real line in terms of coverings by suitable sequences of half-open intervals.
      We will see in Section 3.9 (assuming the axiom of choice) that Lebesgue outer measure is NOT a measure on the set of all subsets of the real line.
    • Part 14 of 15: Slides 56-61
      The sigma-algebra of Lebesgue measurable sets.
      Lebesgue measure λ (on the Lebesgue measurable sets or on the Borel sets).
      Further properties of Lebesgue measure λ and Lebesgue outer measure λ*: correct length for intervals; translation invariance.

  • Section 3.9: A non-measurable set

    Part 15 of 15: Slides 62-65
    Vitali’s example from 1905 of a subset E of [0,1] such that E is not Lebesgue measurable (implicitly using the axiom of choice).