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

I) Outer measure and Lebesgue measure in R^n.

1.1 On the cardinality of infinite sets. Countable sets

1.2 Lebesgue coverings

1.3 Outer measure in R^n

1.4 Measurable subsets of R^n

1.5 Fundamental properties of the Lebesgue measure and of measurable sets

1.6  Limit theorems for "nested" sequences of measurable sets

1.7 Measurability and topology. Sigma-algebras

1.8 Sigma-algebra generated by a family of subsets. The Borel Sigma-Algebra of R^n

1.9 The Borel Sigma-Algebra B(R) of R

1.10 Borel sets, measureable sets and inner measure

1.11 Coordinate transformations and invariance properties of the Lebesgue measure

II) Measurable functions and the  Lebesgue integral

2.1 Measurable functions

2.2 The Riemann integral

2.3 The Lebesgue integral for Simple Functions

2.4 The Lebesgue integral for limited functions with a domain of finite measure

2.5 The Lebesgue integral for non-negative measurable functions

2.6 Summable functions oand the general Lebesgue integral

III) Calculation of measures and integral for domains in R^n: the Fubini-Tonelli theorem and "multiple integrals"

Readings/Bibliography

1) Andrea Brini's lecture notes in Pdf, available on the website

2) H.L. Royden, Real Analysis, The Macmillan Company, 1968

Teaching methods

We will introduce general concepts and methods on the Lebesgue measure and integrals in R^n .

We also analyze some concrete problems, to stimulate the student to find solutions in an independent way.

Assessment methods

The final assessment will consist of an oral examination of 45 minutes. We will assess the student's learning both in terms of concepts and methods, and their ability to apply those to concrete examples.

The student will carefully study five proofs at their own choice
(among the proofs explained in the couse). One of them will be
discussed during the exam.

Office hours

See the website of Nicola Tito Pagani