ICTP Real Analysis 2025

This is the course website for the ICTP course Real Analysis for diploma students during the first semester of 2025. After taking the course, students will be familiar with the Lebesgue measure and the Lebesgue integral and their basic properties. We follow the text books

E. M. Stein and R. Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces. Princeton, NJ: Princeton University Press (2005; Zbl 1081.28001)
and L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions. Revised ed. Boca Raton, FL: CRC Press (2015; Zbl 1310.28001)

Other inspirational sources are

the Real Analysis lectures by Emanuel Carneiro at ICTP, and
the lecture notes by Juha Kinnunen at Aalto University on measure and integral and
on real analysis

Grading

50% exercises, 50% final exam

Lectures and exercise sessions

The lectures take place Tuesdays and Thursdays 14:15-15:45 in the Fibonacci room in the Galileo Guest House.

Lecture notes

main pdf file
other formats: for small screens, for very small screens, landscape
informal overview

The lecture notes will be updated continuously. Below are the latest changes:

Date	Change
2025-10-28	Fix typos from 2025-10-28 and add explanation why the two Radon measures in the proof of Theorem 3.2.3 have the same measurable sets.
2025-10-26	Correct wrong inclusion in sheet 7, exercise 3d to L^p ⊂ L^{p,∞}.
2025-10-26	Add convolution and approximation of identity section.

all changes

Schedule and Exercises

Please read the corresponding sections of the lecture notes before the lecture. We will not cover every detail of the material during the lectures but instead have more interactive sessions during which we focus on the more difficult parts and try to gain a more general understanding of the material.

After having read the material you can test your knowledge by trying to answer the questions from the corresponding quiz.

Please hand in the exercises before Thursdays lecture.

Week	Topics	Sections	Quiz	Quiz solutions	Exercise	Exercise solutions
1	Definition of Lebesgue measure	1.1-1.2 until Theorem 1.2.2 (without proof)	Quiz 1 (pdf), Quiz 1 (html)	Quiz 1 solution (pdf), Quiz 1 solution (html)	Exercise 1 (pdf)	Solution exercise 1
2	Measurable sets	1.2, until Corollary 1.2.18	Quiz 2 (pdf), Quiz 2 (html)	Quiz 2 solution (pdf), Quiz 2 solution (html)	Exercise 2 (pdf)	Solution exercise 2
3	approximation of measurable sets, measurable functions	sections 1.2.3 and 1.3.1	Quiz 3 (pdf), Quiz 3 (html)	Quiz 3 solution (pdf), Quiz 3 solution (html)	Exercise 3 (pdf)
4	approximation of measurable functions and Lebesgue integral of simple functions	sections 1.3.2 and 2.1.1 until Definition 2.1.2	Quiz 4 (pdf), Quiz 4 (html)	Quiz 4 solution (pdf), Quiz 4 solution (html)	Exercise 4 (pdf)	partial solutions exercise 4
5	Lebesgue integral of simple and nonnegative functions (and general measurable functions)	sections 2.1.1 and 2.1.2 (and 2.1.3)	Quiz 5 (pdf), Quiz 5 (html)	Quiz 5 solution (pdf), Quiz 5 solution (html)	Exercise 5 (pdf)
6	L^p-spaces, Fubini's theorem	sections 2.2 and 2.3	Quiz 6 (pdf), Quiz 6 (html)	Quiz 6 solution (pdf), Quiz 6 solution (html)	Exercise 6 (pdf)	Partial solution exercise 6
7	The Lebesgue differentiation theorem and Radon measures on ℝ	sections 3.1 and 3.2			Exercise 7 (pdf)	Partial solution exercise 7
8	Radon measures on ℝ, the Cantor set and functions of bounded variation	sections 3.2, 3.3 and 3.4			Exercise 8 (pdf)	Partial solutions exercise 8
9	Signed measures, convolutions and approximation of the identity	sections 4.1 and 4.2

Solutions exam