TMA4225 Foundations of analysis: Fall term 2014

Jump straight to: This week · Future weeks · Earlier weeks

The plan is to post some information on the lectures here: Both advance information on what I intend to do, and later information on what I actually did (a.k.a. damage control).

We start on Chapter 8, on differentiability and absolute continuity. We shall prove the measure theoretic analogue of the fundamental theorems of calculus.

Dini derivatives, Vitali covering lemma, start on differentiability of monotone functions. See new note “Differentiability” under the Resources link!

Finished differentiability of monotone functions (skipped the proof of “limits to wiggliness” in the notes – I described this proof as just like the previous proof, but on steroids). First fundamental theorem of calculus.

Absolute continuity, second funfamental theorem of calculus, functions of bounded variation.

Summing up, hightlights, whatever I might have forgotten during the term.

I will assume that much of the material of Chapter 1, and of the earlier parts of Chapter 2, are known. But not all of it! I shall go back and cover the missing parts as we go along. I may also pause to revisit these pages when needed.

Rather than just starting on page 1 and going through the book, I made some general remarks about the course, trying to give some idea what it is all about. Then I introduced the Cantor set (section 2.5) and the associated Cantor function. I argued that the Cantor set is a null set without actually telling you in detail what a null set is! But the fact that the Cantor set is contained in a finite union of intervals of arbitrarily small combined lengths should give you the general idea. I even introduced the notion of almost everywhere, a phrase used so much that we like to abbreviate it a.e., to mean everywhere except on a null set.

A historical note: What the book calls the Riemann integral is really the Darboux integral. The Riemann integral is based on Riemann sums defined from tagged partitions, while the Darboux integral is based on upper and lower Darboux sums (“Riemann sums” in the book). This is not very serious, for the Riemann integral and the Darboux integrals are equivalent (they both have the same integrable functions, and the same value for the integral).

First, about summing over infinite sets. (See note on the Resources page.) I spent quite a bit of time on it, because the proof techniques are probably unfamiliar (though not terribly difficult, I hope you agree), and we will need similar techniques later in the course.

Second, about compactness. There is a note about that as well. I forgot to mention the name Heine–Borel for the theorem which states that a closed and bounded subset of the real line is compact.

I did give the definition of Lebesgue outer measure \(\lambda^*\), but did not have time to derive any of its properties.

I had to go to Bergen for a meeting, so Eugenia Malinnikova lectured. She covered the basic properties of Lebesgue outer measure (section 3.2); just enough to give you a proper understanding of null sets (sets \(E\) for which \(\lambda^*(E)=0\)). Further, she started on the Riemann Darboux integral (section 2.6) up to the middle of p. 70.

The plan was to finish section 2.6 on the Darboux integral, in particular showing the main theorem about necessary and sufficient conditions for its existence. In reality, we only got into the middle of this proof, finishing one direction – the one saying that if \(f\) is Darboux integrable, then the set of discontinuity points of \(f\) has measure zero.

Finished the proof of Theorem 2.7 on necessary and sufficient conditions for the Darboux integral to exist. Thus we are done with § 2.6, except for the last bit on convergence properties of the Darboux integral, which we are going to drop (for now at least).

Next, we moved back to § 1.4 on algebras, σ-algebras, and monotone classes. But we did not get much beyond algebras.

Finish § 1.4, including the monotone class theorem.

Then started on § 3.4 with the definition of Lebesgue measurable sets, except I followed my own notes instead of the book. (Notes available under Resources in the left margin. Note that they have been updated twice since I first posted them.) Finished with the proof that intervals are measurable.

Cancelled due to illness

Done with the definition and very basic properties of Lebesgue measure (§ 3.4 or my notes).

Then I covered the example from the end of §3.3, “constructing” a non-measurable set. The reason I write that word in quotes is that we relied on the axiom of choice, hence the existence proof is not at all very constructive. I pointed out that this is unavoidable: One can take standard set theory, throw out the axiom of choice (but keep a countable version), and add other axioms, ending up with a set theory in which all subsets of the real line are Lebesgue measurable. At the end, I said a couple words about the Banach–Tarski paradox.

Start on Ch 4, defininig the Lebesgue integral and deriving its basic properties. There is another note on the resources page.

I was in in Bergen once more, and Eugenia Mallinikova lectured. She covered what was left over in § 4.2, and § 4.3 up to the dominated convergence theorem (DCT), not including its corollaries. (That should be the last of my planned absences this semester.)

Disclaimer: I wrote this up much later, based on my notes. This could be a bit inaccurate.

Started by tying up loose ends: A bit about the extended real numbers \(\mathbb{R}^*=[-\infty,\infty]\) and measurable \(\mathbb{R}^*\)-valued functions, and what it means for such a function to be integrable (one important criterion: it can only take the values \(\pm\infty\) on a set of measure zero). Also a bit about the concept of almost everywhere (a.e.; § 4.4), and the extension of the Dominated convergence theorem (DCT) to sequences that converge a.e..

I continued with some corollaries to the DCT and finished § 4.3.

Covered the leftover pieces of Ch. 4. Primarily, the fact that Darboux/Riemann integrable functions are Lebesgue integrable, and that the Darboux/Riemann and Lebesgue integrals have the same value. In other words, the Lebesgue integral is a true extension of the Darboux/Riemann integral.

Then I went all the way back to § 3.1 and covered it, briefly, in my own way. I have followed the standard definition of Borel set: Borel sets are the members of the Borel σ-algebra \(\mathcal{B}\), which is the σ-algebra generated by the intervals (or open sets). It turns out that the set of Borel measurable functions is precisely the smallest set containing the continuous functions, and that is closed under pointwise sequential limits (i.e., the limits of pointwise convergent sequences of functions). The proof is in parts reminiscent of the proof of the monotone class lemma.

Started with showing the existence of a Lebesgue measurable, non-Borel measurable set. This was really exercise 3.50 on page 109, perhaps not exactly as the authors intended. (Or did I do this on Thursday the previous week? It's possible.)

Then we started on Chapter 5, the abstract Lebesgue integral. I intended to cover quite a bit of material quickly, since it is very similar to the corresponding theory for the Lebesgue integral. Hopefully, this will give us more time for a more relaxed approach to later developments. Accordingly, we covered § 5.1, § 5.2 and a small bit of § 5.3.

The monotone convergence theorem for the abstract Lebesgue integral (in § 5.3), then a whole lot of examples of measure spaces.

Because of all the examples last Thursday, I don't think we are quite done with § 5.3 – so we finish it now, with Fatou's lemma. Then we move on to § 5.4, where we have already defined integrability and the integral for general functions. But we cover the Dominated convergence theorem (DCT), extend the integral (and DCT) to complex valued functions, and then move on to § 5.5 on convergence in measure. Here, at last, is something genuinely new, and not merely a generalization of what we have done for the Lebesgue integral (though the results are applicable to the Lebesque integral as well).

Finished § 5.5 on convergence in measure.

Started on Chapter 6 next, on constructing measures. I departed from the book a bit – see notes on premeausres and Carathéory on the Resources page. We are not yet done with § 6.1.

Finished § 6.1 on constructing measures, but following my notes. The note on premeasures has been updated with a proof of uniqueness of the measure extending the given premeausure, under an assumption of σ-finiteness. My proof uses the monotone classes theorem.

Cointossing space as an example of an abstract measure space – see note in the Resources page.

Lebesgue–Stieltjes measure (§ 6.2, skipping many details).

The final fifteen minutes: a quick look at product measures.

Got going with product measures for real … (§ 6.3): I showed that the measurable rectangles form a semi-algebra with a naturally occuring premeasure on it, and concluded that there is therefore a unique extension to a measure on the σ-algebra generated by the measurable rectangles. We call this the product measure, on the product σ-algebra. (My proof was hopefully quite convincing for the case when both σ-algebras were finite – the extension to general σ-finite σ-algebras was more shaky, but we just ignore that problem and move on.)

I also proved Tonelli's theorem (§ 6.4). It was a quite busy day!

First, Fubini's theorem (§ 6.4), then I did some examples.

I took a sidetrip into Littlewood's principles. From the examples we have seen, you may think that measurable sets and measurable functions (on the real line) can be quite wild, and you're right – but from another perspective, they are quite nice. Littlewood summed it up in three principles:

1. Any measurable set is almost a finite union of intervals
2. Every measurable function is almost continuous (Lusin's theorem)
3. Pointwise convergence is almost uniform (Egorov's theorem)

I have taken some inspiration from a recent paper by Rolando Magnanini and Giorgio Poggesi. But I will write up a note – there just hasn't been time to do so yet.