85161 - Measure Theory

Learning outcomes

By the end of the course the student is familiar with the basic concepts and results of Lebesgue measure theory (outer measure, measurable sets and connections with topology, Borel sigma algebra) as well as of Lebesgue theory of integrals (measurable functions/random variables, convergence theorems, the Fubini/Tonelli theorem for multivariate integration).

Course contents

1. The measurement problem: Cells and boxes in Rn, volume, coverings, and the concept of outer measure.

2. Outer measure m*: Definition, monotonicity, countable subadditivity, and translation invariance.

3. Caratheodory measurability: Measurable sets, Lebesgue measure, and measure extension.

4. Basic properties of Lebesgue measure: Continuity from below and above; nested sequences of measurable sets.

5. Topology and Borel sets: Generated sigma-algebras, Borel sets, and the relation between Borel and Lebesgue sets.

6. Regularity and inner measure: Approximation by open, closed, and compact sets; invariance properties.

7. Measurable functions: Definition, stability properties, simple functions, and approximation.

8. Lebesgue integral I: Integral of simple functions and definition for nonnegative measurable functions.

9. Convergence theorems I: Monotone Convergence Theorem, Fatou's Lemma, and examples.

10. Riemann vs Lebesgue: Motivation, comparative advantages, and selection of convergence theorems.

11. Lebesgue integral II: Integrable functions, functional decomposition, and introduction to L1 spaces.

12. Convergence theorems II: Dominated Convergence Theorem and its applications.

13. Product measures: Rectangles, sections, construction principles, and setup for Tonelli and Fubini theorems.

14. Tonelli's theorem: Iterated integrals for nonnegative functions and multivariate examples.

15. Fubini's theorem: Iterated integrals for L1 functions and final applications.

Readings/Bibliography

The textbook used in the course is

BARTLE, Robert G. The Elements of Integration and Lebesgue Measure. New York: Wiley-Interscience, 1995. (Wiley Classics Library).

A more advanced and useful Bibliography for reference is

ROYDEN, Halsey L.; FITZPATRICK, Patrick. Real Analysis. 4. ed. Pearson, 2010.

Teaching methods

The course focuses on the construction of the Lebesgue measure in Rn as the fundamental reference for introducing the general theory of measure and integration.

We will present the formal definitions and the main convergence theorems, providing the necessary grounds for abstract integration. The methods include lectures on theoretical concepts and the discussion of concrete problems to help students develop the technical rigor required for independent analysis.

Assessment methods

Assessment Methods

The assessment consists of a written examination followed by an oral examination.

1. Written Examination:
The written test consists of 6 problems on the theoretical topics of the course, each valued at 10 points (totaling 60 points). This score is then explicitly converted to the 0-30 scale used by the University (by dividing the total points by 2). A minimum score of 18/30 is required to pass the written exam and qualify for the oral phase.

2. Oral Examination:
The oral exam consists of 3 specific questions:
- Question 1: A conceptual question regarding definitions or fundamental properties.
- Question 2: Reproduction or discussion of a solution from a problem in the written exam (selected by the instructor).
- Question 3: A simple exercise on a course topic OR a discussion regarding the main idea of a proof or example presented during the lectures.

3. Final Grade:
The final grade is calculated as the sum of the written score (on the 0-30 scale) and the points obtained in the oral exam (up to a maximum of 3 points). The oral examination will not decrease the score obtained in the written exam.
If the final sum exceeds 30 points, the student will be awarded "30 cum laude" (30 e lode).

Representative problem sets will be posted weekly on the Virtuale platform. These sets, along with examples discussed in class, serve as the reference guide for the types of problems expected in the written and oral examinations. Students are welcome to attend office hours to discuss these problems or clarify any questions.

Office hours

See the website of Diego Ribeiro Moreira