Course Title: THEORY OF INTEGRATION
Level: Graduate
Course Type: Elective
No. of Credits: 4
Pre-requisites: None
Course Rationale
The Riemann integral is inadequate in analysis, because many functions, arising naturally from limiting processes, cannot be Riemann integrated. At the turn of the century, Lebesgue devised a method for integrating any function one is likely to encounter. Furthermore this method extends with no additional effort to more general settings. This course develops this essential tool and gives a few of its applications.
Any student intending to do advanced analysis, statistics, applied or pure mathematics; in particular, those interested in probability theory, differential equations, harmonic analysis, ergodic theory, functional analysis, dynamical systems. This course is very important for those interested in finance.
Course Description
In this course, we consider the limitations of the Riemann integral, and show that it is necessary to develop a precise mathematical notion of ‘length’ and ‘area’ in order to overcome these deficiencies. In so doing, we create a precise concept of measure, and use it to construct the more powerful Lebesgue integral. Finally we look at applications of measure and Lebesgue integration in modern probability theory.
Students will be expected to have a solid background in undergraduate calculus and real analysis.
Learning Outcomes
Upon successful completion of the course, students will be able to:
- Provide a first principles construction of the Riemann Integral
- Demonstrate the properties of Riemann integrable functions
- Illustrate the concept of Lebesgue measure
- Distinguish between countable and uncountable sets, as well as the properties of Lebesgue measure
- Provide a suitable definition of a measurable function on R.
- Perform algebraic operations on measurable and equivalent functions
- Define and outline the basic properties of Lebesgue integral on bounded sets
- Identify and utilize the Monotone Convergence Theorem, Fatou's Lemma; the Dominated Convergence Theorem
- Perform integration over sets of infinite measure
- Provide a comparative analysis of Riemann and Lebesgue integrals and of the generalization of the Fundamental Theorem of Calculus.
- Apply the principles of measure theory to probability theory: recognizing random variables as measurable functions
Content (48 hours)
- Background and motivation: First principles construction of the Riemann integral;
Riemann integrable functions; the role of the Fundamental Theorem of Calculus; the Dirichlet function. (8 hours)
- Lebesgue measure: Measure on R and R2; countable and uncountable sets; properties of Lebesgue measure; construction of non-measurable sets. (12 hours)
- Measurable functions: Measurable functions on R; simple functions; algebraic operations on measurable functions; equivalent functions. (8 hours)
- The Lebesgue integral: Definition and basic properties on bounded sets; the Monotone Convergence Theorem; Fatou's Lemma; the Dominated Convergence Theorem; integration over sets of infinite measure; comparative analysis of Riemann and Lebesgue integrals; generalization of the Fundamental Theorem of Calculus. (12 hours)
- Application to probability theory: Probability spaces as measure spaces; random variables as measurable functions; expectation as Lebesgue integral with respect to a probability measure. (8 hours)
Teaching Methodology
Lectures: Three (3) lectures each week (50 minutes each).
Tutorials: One (1) weekly tutorial session (50 minutes of problem solving, based on theory
covered during lectures).
Assessment
Coursework Mark: 40% - Two 15% Coursework examinations and 10% Assignments (*based on four assignments given during the semester)
Final Examination: 60% (one 3-hour written paper)
Course Calendar
Week |
Lecture/Tutorial Topic |
Assignments |
---|---|---|
1 |
Background and motivation: First principles construction of the Riemann integral. |
- |
2 |
Background and motivation: Riemann integrable functions; the role of the Fundamental Theorem of Calculus. |
Assignment #1 |
3 |
Lebesgue measure: Measure on R and R2 |
- |
4 |
Lebesgue measure: Countable and uncountable sets; properties of Lebesgue measure |
Assignment #2 |
5 |
Lebesgue measure: construction of non-measurable sets |
- |
6 |
Measurable functions: Measurable functions on R; simple functions. |
Coursework Exam #1 |
7 |
Measurable functions: Algebraic operations on measurable functions; equivalent functions. |
- |
8 |
The Lebesgue integral: Definition and basic properties on bounded sets; the Monotone Convergence Theorem. |
Assignment #3 |
9 |
The Lebesgue integral: Fatou's Lemma; the Dominated Convergence Theorem; integration over sets of infinite measure. |
- |
10 |
The Lebesgue integral: Comparative analysis of Riemann and Lebesgue integrals; generalization of the Fundamental Theorem of Calculus. |
Coursework Exam #2 |
11 |
Application to probability theory: Probability spaces as measure spaces; random variables as measurable functions |
Assignment #4 |
12 |
Application to probability theory: expectation as Lebesgue integral with respect to a probability measure. |
- |
13 |
Revision |
- |
Suggested Texts / References
Text Books:
- H. L. Royden, Real Analysis, Macmillan Publishing Company, N. Y, 1987.
- G. F. Simmons, Introduction to Topology and Modern Analysis, Krieger Publishing Company, 2003.
Reference Books:
- I. K. Rana, An Introduction to Measure and integration, Narosa Book Company, 1997.
- P.K. Jain & V. P. Gupta, Lebesgue Measure and Integration, Wiley Eastern Limited, 1986.
- P.R. Halmos, Measure Theory, Van Nostrand, 1974.
- W. Rudin, Principles of Mathematical Analysis (3rd Edition), McGraw-Hill Book Company, 1976.
- G.de Barra, Measure theory and Integration (2nd Edition) Horwood Publishing Limited, 2003.