The course is mainly based on the Finnish lecture notes "Moderni reaalianalyysi" by Ilkka Holopianen, MoRA.pdf.
Two English versions are available: (a) a translation made at the University of Turku, MoRAeng.pdf, covers Chapters 1-4; (b) an adaptation of the lecturer, mora-eng.pdf, covers most but not all of Chapters 1-4, but also the whole Chapter 5.
There is also an appendix with some additional material not contained in the other two files.
1st exercise set for Jan 18
2nd exercise set for Jan 25
3rd exercise set for Feb 1
4th exercise set for Feb 8
5th exercise set for Feb 22
6th exercise set for Mar 1
Note: In Ex. 6.2, the upper bound for the Hausdorff measure of a product set is not true in the stated generality, but it is true for instance when K, L are the same Cantor sets. This is enough for the claim concerning existence of sets of a given Hausdorff dimension.
7th exercise set for Mar 15
8th exercise set for Mar 22
9th exercise set for Apr 5
10th exercise set for Apr 12
Corrections: Ex. 10.2: The task should be estimating the lower s-density, not lower 1-density. Ex. 10.6: Add the assumption that E has finite Hausdorff 1-meausure.
11th exercise set for Apr 19
Ex 11.2 is not correct as stated. A correct statement to prove is obtained by assuming that the range of mu is [0,1] instead of [0,infinity]. This has been corrected in Lemma A.3 of the notes on Apr 18, and this version of the lemma is all that is needed for Theorem A.4.
The course exam will be held on Monday 30 April at 9:15-12:00 in Exactum B121. (The usual Monday lecture room, but observe that we start one hour earlier!) The actual examination time is 2 h 30 min, the additional 15 min is reserved for settling down in the beginning and returning the papers in the end. The examination will start as soon as we settle down, so be there in time!
Master's Programme in Mathematics and Statistics is responsible for the course.
Belongs to the Mathematics and Applied mathematics module.
The course is available to students from other degree programmes.
Measure and integral, Real analysis I
Advanced knowledge on real analysis, incl. fractals.
Recommended time/stage of studies for completion: 1. or 2. year
Term/teaching period when the course will be offered: varying
Hausdorff measure and dimension, self similar fractals, differentiation of Radon measures, Radon-Nikodym derivative, signed measures.
Required: Moderni reaalianalyysi, luentomoniste.
Recommended: Evans-Gariepy: Measure theory and fine properties of functions. K. Falconer: The Geometry of Fractal Sets.
Lectures and exercise classes.
Exam and excercises, Course will be graded with grades 1-5.
Exam, other methods will be described later