# Measure Theory

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

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 have had a lot of “fun” with attempting to record additional webcam footage of myself. See recent posts in my blog such as
https://explainingmaths.wordpress.com/2012/02/15/camtasia-problems/

for example. However, I am quite impressed by the webcam’s “follow-my-face” software, when it is working! (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.

## 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 failed (see below). I hope to narrate additional screencasts if and when I have time to cover the material on those slides.

• ### Section 3.1, The Extended Real Line

• Slides 20-23: streaming video (requires flash)
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.
• Slides 23-34: streaming video (requires flash)
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

• Slides 25-28: streaming video (requires flash)
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

• Slides 29-30: streaming video (requires flash)
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

• Slides 31-35: streaming video (requires flash)
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.
• Slides 35-36: New in 2017!  Although the original recording failed, I have now (finally) used Camtasia on my Windows Surface to add narration (and digital pointing)  to the annotated slides: video now available on the University of Nottingham’s MediaSpace server.
• ### Section 3.5: The sigma-algebra generated by a collection of subsets

• Slide 37: New in 2017! As above, although the original recording failed, I have used Camtasia to add narration (and digital pointing)  to the annotated slides: video now available on the University of Nottingham’s MediaSpace server.
• Slides 37-40 (notable for the first successful use of my webcam’s follow-my-face software in a lecture!): streaming video (requires flash)
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

• Slides 41-43: streaming video (requires flash)
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.
• Slides 41-47: streaming video (requires flash)
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

• Slides 48-53: streaming video (requires flash)
Deduction of standard properties of measures from the two  axioms: finite additivity; monotonicity; countable subadditivity.
Continuity properties of measures stated and discussed.
• Slides 52-53: streaming video (requires flash)
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

• Slides 54-56: streaming video (requires flash)
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.
• Slides 56-61: streaming video (requires flash)
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

Slides 62-65: streaming video (requires flash)
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).

### 3 responses to “Measure Theory”

1. Very very awesome!! Just watched them all. And I want moar!

Liked by 1 person