Lectures

Finish the monotone class theorem; fix the glitch in my proof on the 2-D Lebesgue measure; and move on to the Tonelli and Fubini theorems (252).

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(… to be filled in …)

(The summary for the first five weeks is a bit inaccurate, especially the references.)

References to Fremlin's book in bold, starting with a digit. Example: 114A.
References to the notes by Eugenia Malinnikova also in bold, starting with the letter N. Example: N3.1.

This was the first lecture, in which I got to ramble on about what is the point of this class. Since this course has the word “foundation” in its title, I felt a need to say a few words about the foundations of mathematics itself, before we even get to the foundations of analysis. (But don't worry; this stuff will not be on any test!) To make a short summary even shorter, mathematics is conventionally based on Zermelo–Fraenkel set theory including the axiom of choice, briefly known as ZFC. (This is not the only possible foundation of mathematics, however, just the most common – for those who even care.) Within the ZFC framework, we take the axiomatic approach to understanding the real number system, but I could not resist briefly mentioning one way to construct the real numbers based on the rational numbers, using Dedekind cuts.

Anyhow, the central axiom of the real number system is the axiom of completeness, here given in the formulation that any nonempty, upper bounded set of real numbers has a least upper bound – a supremum. (There are other, equivalent, formulations, such as the convergence of all Cauchy sequences.)

In the end I proved the Bolzano–Weierstrass theorem, except I made some mistakes at the end as we ran out of time.

After the lecture, I decided to dust off an old note I had lying around, which explains my peculiar notion of subsequences better. I added a proof of the Bolzano–Weierstrass theorem as an example, so you can read about it there. The title of the note is: Diagonals without tears.

I started by going through the note mentioned above, clearing up the notion of subsequences, then giving the diagonal lemma and a proof of the Bolzano–Weierstrass theorem. I gave a proof of the uncountability of the real numbers (really, [0,1]) based on the Bolzano–Weierstrass theorem and a very rudimentary result from measure theory, namely that if a finite set of intervals cover [0,1] then the sum of the lengths of those intervals must be at least 1. (See Fremlin 114B.) Then I defined open, closed and compact subsets of the reals, and got most of the way through Theorem 2.8 of the notes (part 1).

Finished section 2 of the notes, then went back an started on section 1 in earnest, defining the Riemann integral and getting as far as proving Lemma 1.2.

We made it through Riemann's integrability criterion, and showed that continuous functions are integrable.

Next I introduced outer measure with the aim of showing that a function is Riemann integrable if and only if the set of discontinuities of the function has outer measure zero.

Write \(\lambda I\) for the length of an interval \(I\). Then the outer measure of a set \(A\subseteq\mathbb{R}\) is \[ \theta A=\inf\Bigl\{\sum_{k=1}^\infty\lambda I_k\colon \langle I_k\rangle_{\{k\in\mathbb{R}\}} \text{ is a sequence of intervals with } A\subseteq\bigcup_{k=1}^\infty I_k\Bigr\} \] Fundamental properties (which define the general notion of outer measure):

1. \(\theta\emptyset\)=0
2. If \(A\subseteq B\) then \(\theta A\le\theta B\)
3. \(\displaystyle\theta\Bigl(\bigcup_{n=1}^\infty A_n\Bigr)\le\sum _{n=1}^\infty\theta A_n\)

I showed that the outer measure does not change if we restrict the intervals in the definition above to be open, and that \(\theta I=\lambda I\) for any interval \(I\).

And speaking of intervals, here is a concise definition of interval: It is a subset of the real line so that, given any two points in the set, any point between those two also belongs to the set. You should verify that this corresponds to the definition you are used to. Notice that this makes any singleton set an interval, though. Also the empty set is an interval according to this definition. You might not be used to that.

For convenience, I use a couple of concepts and notation that are not in the book:

The variation of a function f over a set A is \[ \mathop{\mathrm{var}}(f;A)=\sup\{f(x)\colon x\in A\}-\inf\{f(x)\colon x\in A\}. \] and the strength of discontinuity of f at a point x is \[ \sigma(f;x)=\inf\{\epsilon>0\colon \exists\delta>0\;\forall y\colon |x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon\} \] (A serious misprint in the above definition has been fixed. I did not get to use the second definition. Next week …)

I proved that a function is Riemann integrable if the set of its discontinuitites has outer measure zero. (The proof that this is necessary is left for the exercises.)

Then I returned to the notes, section 3, to discuss uniform convergence and its relevance to the Riemann integral. I plan to skip the discussion of the Riemann integral in higher dimensions.

Section 4 with the Riemann–Stieltjes integral and functions of bounded variation.

We started on Fremlin's book. The first topic is σ-algebras, which may seem abstract, but I did my best to explain what they are for as well as exploring their basic properties. I also defined the notions of generated σ-algebras and Borel sets.

Measures and their basic properties.

Outer measures and Carathéodory's construction (113). If the Καραθεοδωρή construction seems all Greek to you, there are obvious reasons for that. But do not despair: It is easier than it looks at first glance if you just study it a carefully. I wrote up a central part of the argument in a somewhat different style than the book's. I will follow this note in the lecture: Two A5 pages for screen reading, or one A4 page for printing.

Lebesgue measure (114).

A bit more on the Lebesgue measure on the real line (114). I added a result known as outer regularity:

(i) Given any Lebesgue measurable set \(E\) and \(\varepsilon\gt0\), there exists an open set \(V\supseteq E\) with \(\mu(V\setminus E)\lt\varepsilon\). Moreover,
(ii) there exists a countable intersection of open sets, \(W\supseteq E\), with \(\mu(W\setminus E)=0\).

If this result is not in the book, we'll need to add it to the curriculum.

We will skip the section on higher-dimensional Lebesgue measure (115) – and return to it later.

We start on integration theory (Ch 12). This lecture will be spent inside 121, measurable functions. The point being, the measurable functions will turn out to be the functions we may hope to be able to integrate. The only remaining requirement for integrability will have to do with avoiding infinities.

There was no lecture on 30 September.

Mostly section 122, definition of the integral.

Continued in section 122. This is rather technical, so it takes time.

Finish the definition of the integral (122) and start on the convergence theorems (123). This is a deceptively short, but very important chapter!

Convergence theorems (123): Finished up the proof of B. Levi's theorem, then showed Fatou's lemma and Lebesgue's dominated convergence theorem and the application to differentiation under the integral sign. I also showed a simple example demonstrating what could happen if you drop the “dominated” assumption from Lebesgue's theorem: the functions \(n\chi[0,n^{-1}]\) converge to 0 a.e. but their integrals all equal 1.

Before getting into the lecture I repaired a broken solution to 121Y(e) from yesterday's exercises. My (incomplete!) handwritten note on this can be found on the exercise page.

We start on chapter 22 in volume 2. First Vitali's lemma (221), then differentiating an indefinite integral (222).

Got as far as 222C.

About the midterm test, and try to finish section 222 (excluding the starred sections 222J–L).

No lecture; midterm test instead.

Lebesgue's density theorems (223). Also some smaller items, such as two items that deserve being lifted out from the exercises: Egorov's theorem (131Ya, also known as Littlewood's third principle).

Lusin's theorem, also known as Littlewood's second principle. I have written up a short note on Littlewood's three principles (a5 (for the screen), a4 (for paper)). I made the final part of the proof of Lusin's theorem unnecessarily complicated in the lecture; see the notes. Absolutely continuous functions (225 A–E). I almost finished the proof of 225D, just mopping up remained. I had already showed that the main result (225E) follows.

At the end I presented a car race puzzle (solution to be revealed later).

Finished 225D and 225F. Presented a solution to the car race puzzle, which includes a bit on Cantor sets (both the usual one and generalized). (My handscribbled “slides” probably won't make much sense unless you were there.) A few odds and ends, sometimes skipping a bit of detail: Measurable subspaces (131), outer measure from a measure (132A–C), infinite integrals, generalizations of the integral (133A–G).

We move on to function spaces, selectively picking bits of 241 and 242 for consideration. Next: product measures, iterated integration and Fubini's theorem (251 and 252). But I plan to keep it simple by mostly sticking with σ-finite measure spaces. (… Details to be filled in …)

Completeness of \(L_1\). Product measures: Material carefully extracted from 251 and 252 and simplified appropriately.

Product measures (still), and 2-dimensional Lebesgue measure as a product of 1-dimensional Lebesgue measures. (I hit a snag in a proof that I will correct next week.) Started on the Monotone class theorem (136A–B).