Measure Theory And Integration
- Sciences
- 400 level
- 3 credit units
- 116 pages
- 12 units
This course introduces measure theory and integration, providing the conceptual framework for probability and analysis. It covers measurable spaces, measure spaces, and measurable functions. Students will learn about Lebesgue integration on general measure spaces and the real line. Topics include algebras, sigma-algebras, Banach spaces, product measures, and Fubini's theorem. The course aims to equip students with the necessary foundation for advanced studies in mathematics.
About this course
- Difficulty
- Advanced
- Study hours
- 150 hours
- Maths
- Advanced
- Content
- Theoretical
- Practical work
- No
- Real Analysis
- Calculus
- Set Theory
- Assignments
- Tutor marked assignments
- Final examination
What you'll read
The real module and unit structure of MTH411, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH411 · UNIT2: Fubini’s Theorem
The Presentation Schedule included in your course materials gives you the important dates for the completion of tutor marked assignments and attending tutorials. Remember, you are required to submit all your assignments by the due date. You should guard against lagging behind in your work.
What you should be able to do
- Explain the concepts of measurable space, measure space, and measurable functions.
- Apply properties of measures and measurable functions in integration theory.
- Understand Lebesgue integration for general measure spaces and the real line.
- Apply the theory of integration and convergence to evaluate integrals.
- Describe the space of integrable functions as a Banach space.
- Apply Fubini's theorem to evaluate integrals over product measures and product spaces.
What it prepares you for
- Data Scientist
- Financial Analyst
- Statistician
- Research Mathematician
- Actuary
- Finance
- Data Analysis
- Actuarial Science
- Research
- Academia
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 3: Theory of Integration
Unit 1: Integration of Positive Functions
Requires understanding of limits, sequences, and series, and the ability to apply these concepts in abstract settings.
- Module 3: Theory of Integration
Unit 2: Integration of Complex Functions
Requires a solid understanding of complex numbers and their properties, as well as familiarity with complex analysis concepts.
A suggested way through it
13 weeks, about 105 hours in total. Yours will differ.
- Week 1Module 1: Lebesgue Measure of Subset of ℝ
Unit 1: Measure of a Bounded Open Set · 5 hours
Read the introduction to measure of bounded open sets.. Understand the definition and basic properties of measure of bounded open sets.. Attempt the tutor marked assignment..
- Week 2Module 1: Lebesgue Measure of Subset of ℝ
Unit 2: Measure of a Bounded Closed Set · 5 hours
Understand the definition and basic properties of measure of bounded closed sets.. Attempt the tutor marked assignment..
- Week 3Module 1: Lebesgue Measure of Subset of ℝ
Unit 3: The Outer and Inner Measures of Bounded Sets · 5 hours
Understand the definitions of outer and inner measures of bounded sets.. Attempt the tutor marked assignment..
- Week 4Module 2: General Measure Space (X, M, Jl)
Unit 1: Algebras and Sigma-Algebras · 10 hours
Understand the definitions of algebras and sigma-algebras.. Attempt the tutor marked assignment..
- Week 5Module 2: General Measure Space (X, M, Jl)
Unit 1: Algebras and Sigma-Algebras · 5 hours
Continue studying Algebras and Sigma-Algebras. Attempt the tutor marked assignment..
- Week 6Module 2: General Measure Space (X, M, Jl)
Unit 2: Measures · 10 hours
Understand the definition of measures and their basic properties.. Attempt the tutor marked assignment..
- Week 7Module 2: General Measure Space (X, M, Jl)
Unit 3: Measurable Functions · 10 hours
Understand the definition of measurable functions and their basic properties.. Attempt the tutor marked assignment..
- Week 8Module 3: Theory of Integration
Unit 1: Integration of Positive Functions · 10 hours
Understand the definition of Lebesgue integral of positive functions.. Study the Monotone Convergence Theorem.. Attempt the tutor marked assignment..
- Week 9Module 3: Theory of Integration
Unit 1: Integration of Positive Functions · 5 hours
Continue studying Integration of Positive Functions. Attempt the tutor marked assignment..
- Week 10Module 3: Theory of Integration
Unit 2: Integration of Complex Functions · 10 hours
Understand the definition of Lebesgue integral of complex functions.. Study the Dominated Convergence Theorem.. Attempt the tutor marked assignment..
- Week 11Module 3: Theory of Integration
Unit 3: Lebesgue Integration of Real – Valued Functions Defined on ℝ n · 10 hours
Understand Lebesgue integration of real-valued functions defined on ℝn.. Attempt the tutor marked assignment..
- Week 12Module 4: Classical Banach Spaces
Unit 1: Sets of Measure Zero · 10 hours
Understand the concept of sets of measure zero.. Attempt the tutor marked assignment..
- Week 13Module 4: Classical Banach Spaces
Unit 2: L p – Spaces · 10 hours
Understand the definition and properties of Lp – Spaces.. Attempt the tutor marked assignment..
Preparing for the exam
- Thoroughly review all definitions and theorems related to measure theory and integration.
- Practice solving problems from the textbook and assignments, focusing on applying theorems.
- Create concept maps linking different modules, especially Module 2 (General Measure Space) and Module 3 (Theory of Integration).
- Focus on understanding the conditions and applications of the Monotone Convergence Theorem, Dominated Convergence Theorem, and Fubini's Theorem.
- Review and understand the proofs of key theorems, as they often provide insights into the underlying concepts.
- Allocate sufficient time for revision, focusing on areas where you encountered difficulties during the course.
- Practice past examination papers to get familiar with the exam format and question types.
Questions students ask about this course
What is MTH411 about?
This course introduces measure theory and integration, providing the conceptual framework for probability and analysis. It covers measurable spaces, measure spaces, and measurable functions. Students will learn about Lebesgue integration on general measure spaces and the real line. Topics include algebras, sigma-algebras, Banach spaces, product measures, and Fubini's theorem. The course aims to equip students with the necessary foundation for advanced studies in mathematics.
How many units does MTH411 have?
MTH411, Measure Theory And Integration, has 12 units across 5 modules, over 116 pages of course material. You can read it one unit at a time.
How many credit units is MTH411?
MTH411 carries 3 credit units, at 400 level in Sciences.
Is MTH411 hard?
MTH411 is rated advanced level, with advanced mathematical content. It is mostly theoretical work.
How long does MTH411 take to study?
About 150 hours of study, spread across its 12 units.
How is MTH411 assessed?
MTH411 is assessed by assignments, tutor marked assignments and final examination.
What do I need before starting MTH411?
Real Analysis Calculus Set Theory
What can I do with MTH411?
Data Scientist, Financial Analyst, Statistician, Research Mathematician and Actuary.