Real Analysis, Fall 2023

News:

• NOTE: small changes in presentation schedules (see below). On Thursday Nov 23 we start at 9.00.

• 10th solutions online.

• 9th solutions online.

• 10th (last) exercises and 8th solutions online.

• 9th (and second to last) exercises and 7th solutions online. 

• 8th exercises and 6th solutions online.  

• New set of lecture notes for the Hardy-Littlewood maximal function, which more closely match the lecture on Oct 19.

• Fifth solutions and 7th exertices online.

• Sixth exercises online.

• Fourth solutions online.

• Fifth exercises online.

• Uploaded a new set of lecture notes for Fubini's theorem, which more closely match the lecture on Sept 29. 

• Dates for all presentations: Nov 23-24, Nov 30 and Dec 1. Solutions 3 online. 

• Typo corrected in the 3rd exercise sheet (in the notion of Borel regularity, the set B needs to contain A)

• List of possible presentation topics online (see right above exercises).

• Fourth exercises and Solutions 1-2 online

• Third exercises online.

• Second exercises online.

• First exercises online.

• Passing the course will be based on solving exercises and preparing a (written + oral) presentation of roughly (5 pages + 45 minutes). The grade is based on the exercises. This format may still change if so many students sign up that there isn't enough time for presentations  (in that case we'll have an exam).

• First lecture: Thursday, September 7

• Course website launched. The course will be lectured in English.

Lecture times and exercise sessions:

• Lectures on Thursdays and Fridays 10-12, in MaD 245. 

• Exercises Thursdays 8.30-10.00, in MaD 245. First meeting: September 14

Course description:

The course Real Analysis will pick up where Measure and Integration left off. We will focus less on Lebesgue measure, and more on abstract measures. We will start by proving three key theorems in abstract measure theory, namely

• The Riesz representation theorem (both positive and complex versions)

• Fubini's theorem in sigma-finite measure spaces

• The Radon-Nikodym theorem

After these fundamental results, the following topics are covered:

• Inner and outer regularity properties of Borel measures

• Hardy-Littlewood maximal function and the Lebesgue differentiation theorem

• Smooth approximation of Lp functions (convolutions) 

• Marcinkiewicz interpolation theorem 

• Differentiability of Lipschitz functions (Rademacher's theorem, Stepanov's theorem)

• Basics of Hausdorff measures and densities

• Co-area inequality

• Absolutely continuous functions and functions of bounded variation on the real line

• Whitney's extension and Lusin's approximation theorems

Course materials:

We will be mostly be following Walter Rudin's Real and Complex Analysis. The following hand-written lecture notes are largely base on that book. Acquiring the book isn't necessary, the notes cover everything.

Tuomas' notes:

• Part I (Review of abstract measure and integration theory)

• Part II, Part III (The Riesz representation theorem in locally compact Hausdorff spaces)

• Part IV (Inner and outer regularity of Borel measures, Lebesgue measure via Riesz representation)

• Part V, Part VI (Fubini's theorem in general sigma-finite measure spaces; see also these notes which better match the lecture from Fall '23)

Katrin's notes:

• Radon Nikodym theorem (Part I, Part II, Part III, Part IV)

• Hardy-Littlewood maximal function, and the Lebesgue differentiation theorem (Part I, Part II, Part III, Part IV; see also these notes which better match the '23 lecture)

• Differentiation of measures (Part I, Part II)

• Absolutely continuous functions, and the Fundamental theorem of calculus for Lebesgue measure (Part I, Part II, Part III)

Tuomas's notes

• Rademacher's theorem

• Topics on Hausdorff measures and Lipschitz maps

Katrin's notes

• Whitney's extension theorem (Part I, Part II)

Presentation topics

Schedule:

• Thursday Nov 23: Guangzeng Yi & Reetta Koskelo & Miika Manu

• Friday Nov 24: Lowig Duer & Janne Taipalus

• Thursday Nov 30: Angha Agarwal & Miro Arvila

• Friday Dec 1: Maik Ofensberger & Chengxi Li

The length of the presentation should be 5 p. of written text + 45 minutes of oral presentation. Topics are suggested below, but you'll need to plan yourself how to make the presentation into a "satisfying" piece under the space/time constraints. You probably can't fit the details of all the proofs, but you should write the details of some proofs. Same applies for the oral presentation: you need to assess carefully which details are "the most interesting" for the audience, and which details can be left out.

• BMO spaces and the John-Nirenberg inequality, (Grafakos, Modern Fourier Analysis, S. 7.1) -- reserved, Chengxi Li

• Kirszbraun extension theorem for Lipschitz functions (Heinonen, Lectures on Lipschitz analysis)

• Metric derivative for curves (Ambrosio and Tilli, Topics on Analysis in Metric Spaces, Section 4.1) -- reserved, Janne Taipalus. 

• Kakeya maximal function (Pertti Mattila, Fourier analysis and Hausdorff dimension, Theorem 22.5) -- reserved, Guangzeng Yi

• Dirichlet problem and harmonic measure (these notes of Xavier Tolsa and Marti Prats; this topic may require some background on PDEs) -- reserved, Maik Ofensberger.

• Steiner symmetrisation and isodiametric inequality (Evans and Gariepy, Measure Theory and Fine Properties of Functions, S. 2.2)

• Area formula (Evans and Gariepy, as above, S. 3.3 -- the previous two topics would make a good pair if 2 people are interested) -- Miika Manu

• Besicovitch and Vitali covering theorems (Pertti Mattila, Geometry of Sets and Measures, Theorem 2.8) -- reserved, Lowig Duer

• Lebesgue differentiation in sigma-finite dimensional spaces (Rigot, https://arxiv.org/abs/1802.02069, Section 5, at least Theorem 5.2. The previous 2 topics again make a good pair) -- reserved, Arttu Sorsa

• Equicontinuity, Arzela-Ascoli theorem, normal families of holomorphic functions (Theorems 11.28 and 14.6 in Rudin) -- reserved, Reetta Koskelo

• Tangent measures (Pertti Mattila, Geometry of Sets and Measures, Chapter 14) -- reserved, Miro Arvila.

• Energies, capacities and Frostman's lemma (Xavier Tolsa, Analytic capacity, Theorem 1.23, for Frostman's lemma; Pertti Mattila Geometry of Sets and Measures, p.109-p.112 for energies and capacities) -- reserved, Angha Agarwal

Exercises:

Exercise sessions in the 1st period were hosted by Francesco Nobili. Exercise sessions starting October 19 were hosted by Max Goering.