PURE MTH 3009 - Integration and Analysis III
North Terrace Campus - Semester 2 - 2017
General Course Information
Course Code PURE MTH 3009 Course Integration and Analysis III Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Undergraduate Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Prerequisites MATHS 2100 Course Description 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.
Course Coordinator: Associate Professor Nicholas Buchdahl
The full timetable of all activities for this course can be accessed from Course Planner.
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 examples of these concepts.
3. Prove the basic results of measure theory and integration theory.
4. Demonstrate understanding of the statement and proofs 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 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
Recommended ResourcesH. 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 LearningThis 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 ModesOver the course of 30 lectures, the lecturer presents the 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 will be recorded (when 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 them with the lecturer and their peers. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, allowing them to gauge their progress.
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 SummaryLecture Outline
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 Carathéodory 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.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Component Weighting Outcomes Assessed Assignments 25% All Tutorials 5% All Exam 70% All
Assessment Related RequirementsAn aggregate score of at least 50% is required to pass the course.
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 1% Class exercises 2 Week 3 Week 4 5% Tutorial exercises 3 Week 4 Week 5 1% Class exercises 3 Week 5 Week 6 5% Tutorial exercises 4 Week 6 Week 7 1% Class exercises 4 Week 7 Week 8 5% Tutorial exercises 5 Week 8 Week 9 1% Class exercises 5 Week 9 Week 10 5% Tutorial exercises 6 Week 10 Week 11 1% Class exercise 6 Week 11 Week 12 5% The best five of the six class exercises will count towards the final mark. Final examination Exam period 70%
Assignments will have a maximum two-week turn-around time for feedback to students.
Late assignments will not be accepted.
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.
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