## PURE MTH 7071 - Integration & Analysis

### North Terrace Campus - Semester 2 - 2016

The Riemann integral works well for continuous functions on closed bounded intervals, but it has certain deficiencies that cause problems, for example, in Fourier analysis and in the theory of differential equations. To overcome such deficiencies, a "new and improved" version of the integral was developed around the beginning of the twentieth century, and it is this theory with which this course is concerned. The underlying basis of the theory, measure theory, has important applications not just in analysis but also in the modern theory of probability. Topics covered are: Set theory; Lebesgue outer measure; measurable sets; measurable functions. Integration of measurable functions over measurable sets. Convergence of sequences of functions and their integrals. General measure spaces and product measures. Fubini and Tonelli's theorems. Lp spaces. The Radon-Nikodym theorem. The Riesz representation theorem. Integration and differentiation.

• General Course Information
##### Course Details
Course Code PURE MTH 7071 Integration & Analysis Mathematical Sciences Semester 2 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week Y MATHS 2100 ongoing assessment 30%, exam 70%
##### Course Staff

Course Coordinator: Associate Professor Nicholas Buchdahl

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes

1. Demonstrate understanding of the basic concepts underlying the definition of the general Lebesgue integral.

2. Demonstrate familiarity with a range of these concepts.

3. Prove basic results of measure theory and integration theory.

4. Demonstrate understanding of the statement and proof of the fundamental integral convergence theorems, and their applications.

5. Demonstrate understanding of the statements of the main results on integration on product spaces and an ability to apply these in examples.

6. Apply the theory of the course to solve a variety of problems at an appropriate level of difficulty.

7. Demonstrate skills in communicating mathematics both orally and in writing.

##### University Graduate Attributes

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
1,2,3,4,5,6
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
1,2,3,4,5,6
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
7
• Learning Resources
None.
##### Recommended Resources
H. L. Royden, Real Analysis, 519.53 R8884
W. Rudin, Real and complex analysis, 517 R91r.3
M. E. Taylor, Measure theory and integration, 510.5 G733
##### Online Learning
This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that students make appropriate use of these resources.

• Learning & Teaching Activities
##### Learning & Teaching Modes
Over the course of 30 lectures, the lecturer presents the course material to the students and guides them through it.  During this time students are expected to engage with the material being presented in lectures, identifying any difficulties that may arise in their understanding of it, and interacting with the lecturer to overcome these difficulties.  It is expected that students will attend all lectures, but lectures may be recorded (where facilities allow for this) to help with incidental absences and for revision purposes.  In fortnightly tutorials students present their solutions to assigned exercises and discuss these with the lecturer and their fellow students.  Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
##### Workload

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Activity                                Quantity                           Workload Hours
Lectures                                      30                                         90
Tutorials                                       6                                          18
Assignments                                 6                                          48

TOTAL                                                                                    156
##### Learning Activities Summary
Week 1: Introduction; review of completeness of the real numbers; cardinality; countable and uncountable sets; introduction to measure theory; σ-algebras.

Week 2: Borel sets; the extended real numbers; Lebesgue outer measure and its properties.

Week 3: Lebesgue measurable sets; the σ-algebra of Lebesgue measurable sets; relationship with the Borel σ-algebra; the Cantor set; measure spaces and examples.

Week 4: Properties of measure spaces; measurable functions and their properties.

Week 5: limsup and liminf; sequences of measurable functions; the Cantor ternary function; simple functions; approximation by simple functions.

Week 6: Integration of simple functions; integration of non-negative measurable functions; the Montone Convergence Theorem and its consequences.

Week 7: Fatou's Lemma; the general integral and its properties; the Dominated Convergence Theorem; notions of convergence and Egoroff's Theorem.

Week 8: Comparison of the Riemann and Lebesgue integrals; products of measure spaces.

Week 9: the CarathÂodory Extension Theorem; Tonelli's Theorem and its proof; the Monotone Class Lemma; Fubini's Theorem.

Week 10: Basic concepts of functional analysis: normed vector spaces, Banach spaces and Hilbert spaces.

Week 11: Lp spaces; basic inequalities for Lp spaces; the essential supremum; the Riesz-Fischer Theorem and its proof.

Week 12: Absolutely continuous measures; the Radon-Nikodym Theorem and its proof; the Riesz-Representation Theorem for Lp spaces.

Tutorials in Weeks 3, 5, 7, 9, 11 cover the material of the previous two weeks. A tutorial in Week 1 will review key concepts and results from Real Analysis.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Compontent Weighting Outcomes assessed Assignments 30% All Exam 70% All
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail

Assessment Item                   Distributed                    Due Date                   Weighting

Tutorial Exercises 1                   Week 1                       Week 1                          0%

Class Exercises 1                       Week 1                       Week 2                          5%

Tutorial Exercises 2                   Week 2                       Week 3                          0%

Class Exercises 2                       Week 3                       Week 4                          5%

Tutorial Exercises 3                   Week 4                       Week 5                          0%

Class Exercises 3                      Week 5                        Week 6                          5%
Tutorial Exercises 4                  Week 6                        Week 7                          0%

Class Exercises 4                     Week 7                         Week 8                          5%

Tutorial Exercises 5                 Week 8                         Week 9                          0%

Class Exercises 5                     Week 9                       Week 10                          5%

Tutorial Exercises 6                 Week 10                       Week 10                        0%
Class Exercises 6                    Week 11                       Week 12                        0%
Final examination                         Examination period                                     70%
##### Submission
Assignments will have a maximum two-week turn-around time for feedback to students.

Late assignments will not be accepted.
##### Course Grading

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
Grade Mark Description
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's student’s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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