PURE MTH 3009 - Integration and Analysis III
North Terrace Campus - Semester 2 - 2022
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: Dr Daniel Stevenson
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning Outcomes1. 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)
Attribute 1: Deep discipline knowledge and intellectual breadth
Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.
Attribute 2: Creative and critical thinking, and problem solving
Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.
Attribute 3: Teamwork and communication skills
Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.
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 ModesLecture notes and topic videos will be made available through Mu-Uni. Students are expected to engage with this material every week, attempting quizzes and attending tutorials and workshops to reinforce their understanding of the material. The lecturer will be available to help with weekly consulting sessions and through interaction on the course discussion forum.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload Hours Workshops 11 22 Quizzes 12 12 Tutorials 12 24 Assignments 5 40 Topic Videos ongoing 58 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. Types of convergence. Comparison of the Riemann and Lebesgue integrals.
Week 8: Products of measure spaces. The Carathéodory Extension Theorem.
Week 9: The theorems of Fubini and Tonelli.
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.
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 Tutorial Participation 5% All Quizzes 10% All Assignments 15% All Test 20% All Exam 50% All
Assessment Related RequirementsAn aggregate score of at least 50% is required to pass the course.
Assessment Item Distributed Due Date Weighting Assignment 1 Week 2 Week 3 3% Assignment 2 Week 4 Week 5 3% Assignment 3 Week 6 Week 7 3% Assignment 4 Week 8 Week 9 3% Assignment 5 Week 10 Week 11 3% Test Week 6 Week 6 20% Quizzes Weekly Weekly 10% Final Exam Exam Period 50%
Assignments will have a maximum two-week turn-around time for feedback to students.
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.
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.
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