Here are links to my Measure Theory screencasts from 201112, 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/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 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/camtasiaproblems/
for example. However, I am quite impressed by the webcam’s “followmyface” software, when it is working! (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 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 2023: 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 2334: streaming video (requires flash)
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.
 Slides 2023: streaming video (requires flash)

Section 3.2, Arithmetic and series in the nonnegative extended real numbers
 Slides 2528: streaming video (requires flash)
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.
 Slides 2528: streaming video (requires flash)

Section 3.3, The algebra of limits
 Slides 2930: streaming video (requires flash)
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.
 Slides 2930: streaming video (requires flash)

Section 3.4, Collections of sets
 Slides 3135: 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 sigmaalgebras.  Slides 3536: 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.
 Slides 3135: streaming video (requires flash)

Section 3.5: The sigmaalgebra 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 3740 (notable for the first successful use of my webcam’s followmyface software in a lecture!): streaming video (requires flash)
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.

Section 3.6: Measurable spaces, measurable sets, measures and measure spaces
 Slides 4143: 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 (nonnegative) measure on a sigmaalgebra.  Slides 4147: 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; pointmass measures.
 Slides 4143: streaming video (requires flash)

Section 3.7: Properties of measures
 Slides 4853: 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 5253: streaming video (requires flash)
Proofs of the continuity properties of measures for nested increasing unions and nested decreasing intersections of sequences of measurable sets.
 Slides 4853: streaming video (requires flash)

Section 3.8: Lebesgue outer measure and Lebesgue measure on the real line
 Slides 5456: streaming video (requires flash)
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.  Slides 5661: streaming video (requires flash)
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.
 Slides 5456: streaming video (requires flash)

Section 3.9: A nonmeasurable set
Slides 6265: 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).
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Very very awesome!! Just watched them all. And I want moar!
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