Faculty of Science & Information
Technology
Department: Mathematics
COURSE SYLLABUS
Short Description
Student’s Copy
One copy of this course syllabus is provided to each student registered in this course. It should be kept
secure and retained for future use.
I.
Course Information
1.
2.
3.
4.
5.
Course Title: Real Analysis (Measure Theory and Lebesgue integration)
Course Code: 1309752
Credit Hours: 3
Prerequisite: None
Corequisite: None
II. Instructor Information
1.
2.
3.
4.
Instructor: Dr. Jamila Jawdat
Office:
244 B
Email:
jjawdat@zu.edu.jo
Office Hours: 2 - 3 Sunday and Thursday.
III. Class Time and Place
1. Class Days and Time:
2. Class Location:
1 – 4 pm on Saturdays
303 H
IV. Course Policies
University regulations are applied to this course, regarding Class
Attendance; Punctuality, Exam, Makeup Exams; Absence with permission;
Penalties for Cheating; and Policies for Assignment and Projects. Students
should be aware of all those in addition to other rules and regulations.
V. Resources
Main Reference Text Book:
Real Analysis, by Royden; 4th eddition.
Additional Reference (s):
1. Lebesgue Measure and Integration, by Gupta.
2. Measure theory and integration, by G. de Barra
3. ‫ القياس والتكامل‬:‫ الجزء الثاني‬,‫مبادئ التحليل الحقيقي‬
‫صالح عبدهللا السنوسي‬
VI.
Course Description and Purpose
Course Description: This course is concerned with a generalization of the
Riemann integral (of bounded real functions over bounded intervals) to Lebesgue
integral of measurable functions over measurable sets of R. The course starts with the
concept of outer measure and its properties then proceed to define the Lebesgue
measure on certain sets of R that will be called measurable sets. The later will be studied
along with its properties. Measurable functions over measurable sets will also be
defined and studied. The Lebesgue integral of measurable functions over measurable
sets will be defined along with some properties. Its relation with Riemann integral is
given and certain related theorems will be proved. The course ends with a chapter on
the spaces of measurable and Lebesgue integrable functions, in which some inequalities
are studied that will be used to prove the completeness of these spaces.
VII. Course Learning Outcomes
Upon successful completion of this course, the learner should be able to:
A- Knowledge and understanding (students should):1. Make a good background on basic real analysis and topology.
2. Learn the concept and properties of measure and give some examples
starting with outer measure then the Lebesgue measure.
3. Study measurable sets and measurable functions and their properties.
4. Understand Lebesgue integral and its relation with Riemann integral.
5. Study spaces of measurable Lebesgue integrable functions.
6. Prove some related results and theorems.
B- Intellectual skills and
C- Subject Specific Skills:
Analyze and prove theorems and exercises on each subject that have been studied.
D- Transferable skills – with ability to:Build a good background for more advanced analysis courses like the functional
analysis course and for doing future research in this area.
VIII. Methods of Teaching
The methods of instruction may include, but are not limited to:
1. Lectures
2. Discussion and problem solving
3. Brainstorming
4. Individual assignments
5. Asking students to give a presentation in a specific subject or problem
related to the course
6. Lecturing using PowerPoint Presentations, mixed with discussion with
students
7. Asking students to prepare a term paper about a subject or a problem
related to the course, and discuss it in the class.
IX. Course Learning Assessment/Evaluation
The following methods of learning assessment will be used in this course:
Assessment
a
2 Tests:
Mid Exam
Final Exam
b
Activities such as Quizzes
or Homeworks
c
Assignments Research
proposal
d Presentations/participation
Total
Weight
30%
40%
10%
10%
10%
100%
-
Description
True/False
Short answers
Essay Questions
Problem solving
Explanations
True /False
Short answers
Problem solving
Asking students to
prepare a term paper
about a subject or a
problem related to the
course, and discuss it in
the class
- Student participation
- Course portfolio
X.
Wk
No.
Course Schedule/Calendar
Topic
1&2 Some Revisions
3
4
5
&
6
7
8
9
10
Assignments/
workshops
due date
Reference
in the
textbook
Wk 4
Ch.1
Wk 6
Ch.2
Wk7
Ch.2
Wk 8
Ch.3
Wk 9
Ch.3
Wk 12
Ch.4
Holiday
Lebesgue Outer Measure and its
properties.
Countably subadditive.
The  -Algebra of Lebesgue
Measurable Sets.
Four Characterizations for
Measurable Sets.
Outer and Inner Approximation of
Lebesgue Measurable Sets.
Lebesgue Measure, Countable
Additivity. Non-measurable sets. The
Cantor Set and the Cantor-Lebesgue
Function
Lebesgue Measurable functions.
Sums, Products and Compositions.
Sequential Pointwise Limits and
Simple Approximation.
Egoroff's Theorem and Lusin's
Theorem.
Mid Exam
Review of the Riemann Integral.
Simple Functions. The Lebesgue
Integral of a Bounded Measurable
Function over a Set of Finite Measure.
Properties of the integral. Fatou's
Lemma.
CLO
11
12
&
13
14
&
15
16
Lebesgue Integral of a Measurable
Nonnegative Function. The General
Lebesgue Integral. Countable
Additivity and Continuity of
Integration. Uniform Integrability.
Wk 13
Ch.4
Continuity of Monotone Functions.
Differentiability of Monotone
Functions: Lebesgue Theorem.
Functions of Bounded Variation:
Jordan's Theorem.
Absolutely Continuous Functions.
Integrating Derivatives:
Differentiating Indefinite Integrals.
Wk 14
Ch.6
The Lp Spaces: Definition and
Completeness and their dual.
The Inequalities of Young, Holder,
and Minkowski.
Wk 15
Ch.7
Final Test
V. Special Equipment or Supplies: Computers and internet.