Measure Theory and Real Analysis

Prerequisites:

Fundamental Algebra and Analysis II

Course Description

In this course, students will be first introduced to integration theory by means of Kurzweil-Henstock's integral of functions of one variable and, next, they will be introduced to the power of measure and integration theory as developed by Borel and Lebesgue at the start of the 20^th century. Students will be trained to master these powerful tools in order to be able to apply these subsequently in probability courses, partial differential equations courses, and courses in functional analysis. They will be expected to master both the theoretical aspects (theorems, proofs, applicability) as well as the more technical ones (exercises, calculations, problems).

Learning Objectives

After completing the course, students will be able to:

I. Apply basic tools to compute integrals of functions of one variable: the fundamental theorem of calculus, the change of variable theorem, integration by parts.

II. Understand the role of negligible sets and apply them in calculations.

III. Apply the convergence theorems of Lebesgue's theory and identify cases when they don't apply.

IV. Apply the basic results and understand the basic restriction of derivation of functions of one variable: monotone functions, functions of bounded variation, and absolutely continuous functions.

V. Apply Fubini's and Tonelli's theorems.

VI. Apply the change of variables formula for functions of several variables.

VII. Understand and apply various ways of convergence of sequences of functions.

VIII. Apply classical inequalities: Hölder's, Minkowski's, Chebyshev, Jensen.

IX. Apply the basic tools of Lebesgue spaces: completeness and duality.

X. Apply the Radon-Nikodym theorem and manipulate Radon-Nikodym derivatives, including ways of computing them.

Course Content

I. Part 1: Kurzweil-Henstock integral

• Basics

• Fundamental theorem of calculus

• Change of variable

• Saks-Henstock theorem

• Hake theorem

• Integration over unbounded intervals

• Absolutely integrable functions

• Monotone convergence theorem

• Integrable and measurable sets and their measure

• Covering theorem and consequences

• Negligible sets

• Middle-third Cantor set and Devil's staircase

• Differentiability almost everywhere

II. Part 2: Lebesgue's integration theory

• Measurable spaces and functions

• Basics

• Pointwise sequential limits

• Pointwise sequential approximation by simple functions

• Cavalieri's principle for absolutely KH-integrable functions

• Measures and measure spaces

• Outer measures and Carathéodory's theorem

• Method I

• Countably additive extensions

• Negligible sets and saturated measure spaces

• Integration

• Definition of the Lebesgue integral

• Convergence theorems

• Daniell-Stone theorem

• Product measures

• Basics

• Fubini's and Tonelli's theorems

• Lebesgue spaces

• Basics

• Minkowski's and Hölder's inequalities

• Riesz-Fischer theorem

• Clarkson inequalities and duality of Lebesgue spaces

• Radon-Nikodym

• Absolute continuity

• Signed measures, Hahn and Jordan decompositions

• Radon-Nikodym theorem

• Dual of L₁

• Change of variable theorem

Textbook and Supplementary Readings

1. Lecture notes for the course.

2. D.L. Cohn, Measure Theory.

3. P.R. Halmos, Measure Theory.

4. J.L. Doob, Measure Theory.

5. M.M. Rao, Measure Theory and Integration.