Institution: Umm Al-Qura University

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Course Title:

Measure and integration

Course Code:

^MTH4114-4

Program:

BSc. in Mathematics

Department:

Mathematics

College:

Jamoum University College

Institution:

Umm Al-Qura University

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3 A. Course Identification

1. Credit hours: 4 2. Course type

a. ^University College Department ✔ Others

b. Required ✔ Elective

3. Level/year at which this course is offered: 12 level 4. Pre-requisites for this course (if any):

Real Analysis II (MTH2113-4), Real Analysis I (MTH2112-4) 5. Co-requisites for this course (if any): None

6. Mode of Instruction (mark all that apply)

No Mode of Instruction Contact Hours Percentage

1 Traditional classroom Four

hours/week %100

2 Blended 0 0

3 E-learning 0 0

4 Distance learning 0 0

5 Other 0 0

7. Contact Hours (based on academic semester)

No Activity Contact Hours

1 Lecture 40

2 Laboratory/Studio 0

3 Tutorial 0

4 Others (specify) 0

Total 40

B. Course Objectives and Learning Outcomes

1. Course Description

This module aims to introduce Lebesgue's theory of measure and integration, which extends the familiar notions of volume and "area under a graph" associated with the Riemann integral.

2. Course Main Objective

Measure spaces, measures, outer measures. The Lebesgue measure on 𝑅^𝑛. Measurable functions, the monotone convergence theorem,

Fatou’s Lemma. Integrable functions,

Lebesgue’s dominated convergence theorem and applications.

Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces., transformation formula for the Lebesgue measure on Rn.

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3. Course Learning Outcomes

CLOs Aligned

PLOs 1 Knowledge and Understanding: by the end of this course, the

student is expected to be able to 1.1 Describe the Measure.

1.2 Determine the measurable functions.

1.3 Find out which functions can be integrated, and prove the main properties of the Lebesgue integral.

1.4 Apply and manipulate convergence theorem for the integrals.

2 Skills: by the end of this course, the student is expected to be able to 2.1 calculate different quantities, such as integrals, using convergence

theorems, or Fourier series of simple functions

2.2 Determine whether mathematical objects satisfy certain conditions, such as whether a given function is measurable or integrable;

2.8 Use the concepts and results of the course for proving or disproving statements which the student has not previously seen

C. Course Content

No List of Topics Contact

Hours

1^• Preliminaries 6

2

• Lebesgue measure.

• Measurable functions and their properties. 14

3 Construction and properties of Lebesgue integral.

14 4 Convergence Theorems.

6

Total 40

D. Teaching and Assessment

1. Alignment of Course Learning Outcomes with Teaching Strategies and Assessment Methods

Code Course Learning Outcomes Teaching Strategies Assessment Methods 1.0 Knowledge and Understanding

1.1

At the end of this module students should be able to:

• Understand the construction

Lecture and Tutorials Exams, quizes

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5

Code Course Learning Outcomes Teaching Strategies Assessment Methods and properties of Lebesgue

measure, including the notion and properties of null set;

• Understand the construction of the Lebesgue integral and know its key properties;

• Compute Lebesgue integrals using the Fundamental Theorem of Calculus, Monotone and Dominated Convergence Theorems, and the Tonelli and Fubini Theorems.

coordinates Define the related basic scientific facts, concepts, principles and techniques calculus

2.0 Skills

2.1 ^▪ Ability to apply the measure theory and integration to solve a variety of problems in analysis.

▪ Ability to understand and to develop the statements of the main results in integration and to apply them in examples.

▪ Acquire skills in

communicating mathematics orally as well as in writing.

Lecture/ Individual or group work

Exams, quizzes

3.0 Values

3.1 Prepare for success in disciplines which rely measure theory, and in more advanced mathematics which incorporate these topics

Lecture/ Individual or group work

Exams, quizzes

2. Assessment Tasks for Students

# Assessment task* Week Due Percentage of Total

Assessment Score

1 Midterm Exam 6th %25

2 Quizes and homeworks During semester %25

3 Final exam End of semester %50

*Assessment task (i.e., written test, oral test, oral presentation, group project, essay, etc.)

E. Student Academic Counseling and Support

Arrangements for availability of faculty and teaching staff for individual student consultations and academic advice : -

Each group of students is assigned to a particular faculty where he or she will provide academic advising during specific academic hours. Each staff will provide at least one

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6

session/week. –

There will be an academic advisor how will be a responsible for helping the student by doing the general supervision. –

The people in the library will support the students during the time of the course.

F. Learning Resources and Facilities

1.Learning Resources

Required Textbooks John J. Benedetto, Wojciech Czaja, Integration and Modern Analysis. Birkh?er. 2009.

Essential References Materials

1. Lebesgue Measure and Integration: An introduction, F. Burk 2.Measure, Integral, Derivative: A course on Lebesgue's theory, S.

Ovchinnikov

3. An introduction to classical real analysis, (Karl, R. Stromberg) 4.

Rudin, W.: Real and Complex Analysis, Third Edition, McGraw-Hill Book Company (1987).

Electronic Materials None Other Learning

Materials None

2. Facilities Required

Item Resources

Accommodation

(Classrooms, laboratories, demonstration rooms/labs, etc.)

Classrooms Technology Resources

(AV, data show, Smart Board, software, etc.)

Data Show, Smart Board Other Resources

(Specify, e.g. if specific laboratory equipment is required, list requirements or

attach a list)

None

G. Course Quality Evaluation

Evaluation

Areas/Issues Evaluators Evaluation Methods

Effectiveness of teaching and assessment 

Students Direct

Quality of learning resources Students Direct

Extent of achievement of course learning outcomes

Faculty Member Direct

Evaluation areas (e.g., Effectiveness of teaching and assessment, Extent of achievement of course learning outcomes, Quality of learning resources, etc.)

Evaluators (Students, Faculty, Program Leaders, Peer Reviewer, Others (specify)

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Assessment Methods (Direct, Indirect)

H. Specification Approval Data

Council / Committee Council of the Mathematics Department

Reference No.

Date