Credits: 5

Schedule: 24.02.2020 - 01.04.2020

Contact information for the course (applies in this implementation): 

Instructor: Juha Kinnunen

Assistant: Cintia Pacchiano

Please contact the instructor for questions and comments related to the lectures and the head assistant for the homework assignments.

Teaching Period (valid 01.08.2018-31.07.2020): 

Not lectured (2018-2019)

IV Spring (2019-2020)

Lectured every other year


Learning Outcomes (valid 01.08.2018-31.07.2020): 

After this course you will know how to apply real analysis methods in research.


Content (valid 01.08.2018-31.07.2020): 

Lebesgue spaces (Hölder's and Minkowski's inequalities, Riesz-Fischer theorem, dual spaces and weak convergence), Hardy-Littlewood maximal function (Vitali covering theorem, Marcinkiewicz interpolation theorem, maximal function theorem, Lebesgue's differentiation theorem), convolution approximations, differentiation of Radon measures (Besicovitch covering theorem, Lebesgue points), Radon-Nikodym theorem, Riesz representation theorem, weak convergence and compactness for Radon measures, Sobolev spaces (Poincare and Sobolev inequalities).


Details on the course content (applies in this implementation): 

The course practices will be discussed in the first lecture on Mon 28 October 2019. 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. 

Assessment Methods and Criteria (valid 01.08.2018-31.07.2020): 

Teaching methods: lectures and exercises

Assessment methods: homework assignments and attendance (100%)


Elaboration of the evaluation criteria and methods, and acquainting students with the evaluation (applies in this implementation): 

Grading: There is no final exam. The grading is based on homework assignments and attendance. 

Solved homework assignments      Grade

90%                                                    5

80%                                                    4

70%                                                    3

60%                                                    2

50%                                                    1

Workload (valid 01.08.2018-31.07.2020): 

Contact hours 36 h

Self-study ca 100h


Details on calculating the workload (applies in this implementation): 

Lectures 6*2*2=24 hours. 

Tutorials 6*2=12 hours. 

Independent study of the lecture notes and work on the homework assignments about 100 hours.

Study Material (valid 01.08.2018-31.07.2020): 

All material is available at the course homepage.

Details on the course materials (applies in this implementation): 

The course material consist of lecture notes.

Further reading:

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 

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

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

Substitutes for Courses (valid 01.08.2018-31.07.2020): 



Course Homepage (valid 01.08.2018-31.07.2020):

Prerequisites (valid 01.08.2018-31.07.2020): 

MS-A00XX, MS-A01XX, MS-A02XX, MS-A03XX, MS-A050X, MS-C1350, MS-C1540, MS-E1280


Grading Scale (valid 01.08.2018-31.07.2020): 



Details on the schedule (applies in this implementation): 

The course will run in the teaching period IV. The detailed schedule will be posted to MyCourses during the course.


  1. L^p spaces (Minkowski's inequality, Hölder's inequality, the Riesz-Fischer theorem)
  2. The Hardy-Littlewood maximal function (the maximal function theorems, the Lebesgue differentiation theorem, the Sobolev inequality)
  3. Convolution (approximations of the identity, density of compactly supported smooth functions in L^p spaces, the Poisson kernel)
  4. Differentiation of measures (covering theorems, differentiation theorem for Radon measures, the Radon-Nikodym theorem, the Lebesgue decomposition, Lebesgue points and density points)
  5. The Riesz representation theorem (weak convergence and compactness for Radon measures, weak compacness in L^p spaces, the dual of L^p space)


Registration and further information