lecturer: Piotr Rybka
We mainly follow the book
Measure theory and fine properties of functions by L.C.Evans and R.F.Gariepy.
Any references below are to this text.
02.10 pages 1 - 5.
09.10 Thm. 3 on page 5 -- Thm. 4 on page 9.
16.10 Thm. 5 on page 9, page 11, Thm 2 (Lusin) on page 15.
23.10 Thm 3 (Egoroff) on page 16. Paragraph 1.3, pages 17-22.
30.10 product measure, Fubini Thm.
06.11 Convergence in measure. Equisummability and the Vitali Convergence Theorem.
The Vitali Covering Theorem.
13.11 The Besicovitch Covering Theorem.
15.11 evening mid-term exam from 17:00 till 24:00. Problems will be posted
HERE.
Solutions may be submitted by:
e-mail,
a link to a cloud disk.
File names must contain the FIRST and LAST NAME of the student. I accept only pdf files.
20.11 The Besicovitch Covering Theorem, continued, pages 30-36.
27.11 Differentiation of Radon measures, pages 37-43: Thm 1, Thm 2, Thm 3 from
paragraph 1.6; Lebesgue points, pages 43-45: Thm 1, Cor. 1 and 2.
04.12 no lecture, only problem solving sessions
11.12 The precise representative. The representation of
\((C_c(\mathbb{R}^N; \mathbb{R}^m))^*\),
pages 49-51
18.12 pages 52-55, including Cor. 1; weak convergence and weak compactness for
Radon measures, Thm. 1.
08.01.2020 Definition of the Hausdorff measure \(\mathcal{H}^s\) and its basic properties
15.01.2020 Isodiametric inequality, \(\mathcal{H}^n =\mathcal{L}^n\); Hausdorff dimension.
Real and vector measures.
22.01.2020 The Hahn decomposition theorem. The maximal function, Hardy-Littlewood
inequality.