Real Analysis

News:

• Tenth exercises online. Due on Nov 23.

• Ninth exercises online. Due on Nov 16. 

• Eighth exercises online. Due on Nov 9.

• Seventh exercises online. Due on Nov 2.

• Sixth exercises online (minor fix on Oct 24). Due on Oct 26.

• Fifth exercises online. Due on Oct 19. 

• Fourth exercises online. Due on Oct 12.

• Third exercises online. Due on Oct 5.

• Second exercises are online (since Monday evening, but the last exercise was still swapped on Tuesday morning). Due on Sept 28. 

• First exercises are now online, see below. First exercise session: Sept 21. On Thursday Sept 16 there is no lecture. Instead, watch the 3 first videos (but 3rd video will also be covered in class, so focus on parts 1-2).

• The course is lectured jointly with Katrin Fässler. The teaching assistants will be Damian Dabrowski and Carlos Mudarra. The course is lectured in English (but Tuomas will be happy to answer your questions in Finnish, too). 

• So far we're still planning to have lectures in class (Tuesdays 12-14 and Thursdays 14-16 in MaD380). The same goes for exercises (Tuesdays 14-16, MaD380). The COVID situation may still affect this, so please follow the latest information on this page.

• The first lecture is on Tuesday, Sept 14, at 12.15. 

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

Lecture notes:

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)

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)

• 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)

Video lectures:

There will be pre-recorded videos of a small number of topics. The following are links to Youtube:

• Riesz representation theorem: Part I, Part II, Part III, Part IV, Part V, Part VI

Exercises:

1. First exercises

2. Second exercises

3. Third exercises

4. Fourth exercises

5. Fifth exercises

6. Sixth exercises

7. Seventh exercises

8. Eighth exercises

9. Ninth exercises

10. Tenth exercises