First exercises online.
Course website launched. The course will be lectured in English.
Lectures on Thursdays and Fridays 10-12, in MaD 380. First lecture: September 4.
Exercises Thursdays 8.30-10.00, in MaD 355. First meeting: September 11
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
Measure and integration 1&2
Some topology (you know what a topological space is, and what compactness means in a topological space).
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:
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)
Absolutely continuous functions, and the Fundamental theorem of calculus for Lebesgue measure (Part I, Part II, Part III)
Tuomas's notes
Katrin's notes
The grade for the course is (primarily) based on exercise points, but to pass the course, students will also need to prepare an oral + written presentation (more details in the next section).
One component of passing the course will be an oral + written presentation. 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.
Topics and schedule:
BMO spaces and the John-Nirenberg inequality, (Grafakos, Modern Fourier Analysis, S. 7.1)
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)
Kakeya maximal function (Pertti Mattila, Fourier analysis and Hausdorff dimension, Theorem 22.5)
Dirichlet problem and harmonic measure (these notes of Xavier Tolsa and Marti Prats; this topic may require some background on PDEs)
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)
Besicovitch and Vitali covering theorems (Pertti Mattila, Geometry of Sets and Measures, Theorem 2.8)
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)
Equicontinuity, Arzela-Ascoli theorem, normal families of holomorphic functions (Theorems 11.28 and 14.6 in Rudin)
Tangent measures (Pertti Mattila, Geometry of Sets and Measures, Chapter 14)
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)
First exercises (due Sept 11)
The course assistant for the first half of the course is Matthew Hyde. If you wish to send solutions to exercises by e-mail, you can send them to matthew.j.hyde@jyu.fi.