Here are links to my Measure Theory screencasts from 201112, 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/posHBda for links to a complete set of materials from a full module I gave on Measure and Integration in the years 200678.
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/camtasiaproblems/
for example. However, I was quite impressed by the webcam’s “followmyface” software, when it worked! (See Section 3.5 onwards below.)
Measure Theory
(Chapter 3 from G14FTA Further Topics in Analysis 201112)
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)

Section 3.1, The Extended Real Line

Section 3.2, Arithmetic and series in the nonnegative extended real numbers

Section 3.3, The algebra of limits

Section 3.4, Collections of sets

Section 3.5: The sigmaalgebra generated by a collection of subsets

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

Section 3.7: Properties of measures

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

Section 3.9: A nonmeasurable set
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 preprepared 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 2023
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 2324
Discussion of convergence to infinity for sequences of nonnegative extended real numbers.
Monotone sequences (nondecreasing or nonincreasing) and the Monotone Sequence Theorem for the extended real line.
 Part 1 of 15: Slides 2023

Section 3.2, Arithmetic and series in the nonnegative extended real numbers
 Part 3 of 15: Slides 2528
Addition and multiplication for nonnegative extended real numbers. Problems with the cancellation laws. Series of nonnegative extended real numbers.
Some discussion of problems with the algebra of limits.
 Part 3 of 15: Slides 2528

Section 3.3, The algebra of limits
 Part 4 of 15: Slides 2930
The algebra of limits for sums of sequences of nonnegative extended real numbers.
The algebra of limits for products, and its limitations, for sequences of nonnegative extended real numbers.
 Part 4 of 15: Slides 2930

Section 3.4, Collections of sets
 Part 5 of 15: Slides 3135
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 sigmaalgebras.  Part 6 of 15: Slides 3536
Audio narration, with digital pointer, to go with annotated slides, as the original recording failed
More examples of σalgebras: σalgebras generated by finite collections of sets; sets which are countable or have countable complement; optional discussion of sets with the Property of Baire.
 Part 5 of 15: Slides 3135

Section 3.5: The sigmaalgebra 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 subbase), and how it is harder to give such a concrete approach to generating a σalgebra. The intersection of any family of σalgebras on X is still a σalgebra on X. The definition of the σalgebra 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 3740 (notable for the first successful use of my webcam’s followmyface software in a lecture!)
Properties of the sigmaalgebra on a set X generated by a collection of subsets.
Methods for showing inclusion or equality of sigmaalgebras generated by two different collections of subsets.
The Borel sets in the real line: definition and examples.
 Part 7 of 15: Slide 37

Section 3.6: Measurable spaces, measurable sets, measures and measure spaces
 Part 9 of 15: Slides 4143
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 (nonnegative) measure on a sigmaalgebra.  Part 10 of 15: Slides 4147
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; pointmass measures.
 Part 9 of 15: Slides 4143

Section 3.7: Properties of measures
 Part 11 of 15: Slides 4853
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 5253
Proofs of the continuity properties of measures for nested increasing unions and nested decreasing intersections of sequences of measurable sets.
 Part 11 of 15: Slides 4853

Section 3.8: Lebesgue outer measure and Lebesgue measure on the real line
 Part 13 of 15: Slides 5456
Brief discussion of the following: what we hope to achieve when measuring length, area and volume; the BanachTarski paradox; the Carathéodory extension theorem.
The definition of Lebesgue outer measure λ* on the real line in terms of coverings by suitable sequences of halfopen 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 5661
The sigmaalgebra 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.
 Part 13 of 15: Slides 5456

Section 3.9: A nonmeasurable set
Part 15 of 15: Slides 6265
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).
Very very awesome!! Just watched them all. And I want moar!
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Hi, thanks for these very valuable resources. Headsup, I found a broken link at the following. https://www.maths.nottingham.ac.uk/MathsModules/G13MIN/QuestionSheets.html
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