## Osion kuvaus

• ### Lecture notes

• The course will follow these lecture notes written by Juha Kinnunen. There is no need to buy any books.
• We have adopted a flipped classroom model for the course. The participants are expected to study the announced pages of the lecture notes before each lecture. We shall discuss the material interactively at the lectures.

• Here are some comments about things related to the lecture notes:

• For outer measures, measurability is defined in Definition 1.4, but for measure, measurability just means that the set belongs to the sigma-algebra, where the measure is defined. Usually this does not cause too much confusion as measurable sets for measures also satisfy the condition in Definition 1.4. whenever the quantities are defined.

• #### Schedule:

Mon 26 Oct: Outer measure and measurable sets (1.1-1.2)Thu 29 Oct: Measures and properties of measurable sets (1.3)Mon 2 Nov: Geometric characterizations of measurable sets and metric measures (1.4-1.6)Thu 5 Nov: Lebesgue measure, invariance properties and measurable sets (1.7-1.9)Mon 9 Nov: A Lebesgue nonmeasurable set, Cantor set and definition of measurable functions (1.10-1.11, 2.1-2.2)Thu 12 Nov: Properties of measurable functions, Cantor-Lebesgue function (2.3-2.6)Mon 16 Nov: Approximation of measurable functions, modes of convergence for sequences of functions, Egorov's and Lusin's theorems (2.7-2.9)Thu 19 Nov: The definition of integral and the monotone convergence theorem (3.1-3.3)Mon 23 Nov: Fatou's lemma, integral of a signed function and the dominated convergence theorem (3.4-3.6)Thu 26 Nov: Lebesgue integral and the space of integrable functions (3.7)Mon 30 Nov: Cavalieri's principle and the comparison of Lebesgue and Riemann integrals (3.8-3.9)Thu 3 Dec: Fubini's theorem (3.10-3.11)

• #### Preliminaries

In this course, we assume that the students have studied the course Metric spaces (old name Euklidiset avaruudet / Euclidean spaces). In particular, you should know:

• real numbers, supremum, infimum
• countable and uncountable sets
• open and closed sets, boundary, closure
• compact sets and coverings
• sequences in metric spaces
• function sequences
• continuity

You can study these topics for example from the course material of the Metric space -course or from

Rudin: Principles of Mathematical Analysis

• Section 2 contains most important concepts
• Part of Sections 3 and 4 can be also useful

A.M. Bruckner, J.B. Bruckner, and B.S. Thomson, Real Analysis, Prentice-Hall 1997

L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press 1992

G.B. Folland, Real Rnalysis. Modern Techniques and Their Applications (2nd edition), John Wiley & Sons 1999

F. Jones, Lebesgue Integration on Euclidean Space (revised edition), Jones and Bartlett Publishers 2001

K.L. Kuttler, Modern Real Analysis, CRC Press 1998

W. Rudin, Real and Complex Analysis, McGraw-Hill 1986

E. Stein and R. Sakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press 2005

T. Tao, Introduction to Measure Theory, American Mathematical Society 2011

M.E. Taylor, Measure theory and integration, American Mathematical Society 2006

R.L. Wheeden and A. Zygmund, Measure and Integral: An introduction to Real Analysis, Marcel Dekker 1977

W.P. Ziemer, Modern Real Analysis, PWS Publishing Company 1995

J. Yeh, Real Analysis, Theory of Measure and Integration (2nd edition), World Scientific 2006