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## Course Overview

Measure theory provides a systematic framework to the intuitive concepts of the length of a curve, the area of a surface or the volume of a solid body. It is foundational to modern analysis and other branches of mathematics and physics.

### Course Details

Study Type Level Undergraduate 4 First Sub Session 15 credits (7.5 ECTS credits) Aberdeen No Dr Alexey Sevastyanov

### Qualification Prerequisites

• Programme Level 4

None.

None.

No

### Course Description

1. Lebesgue extension of a measure;Algebras and σ-algebras of sets;Measure spaces;Properties of the Lebesgue measure;
2. Lebesgue extension of a measure defined on a ring of sets;
4. Extension a measure from a semi-ring of sets to the corresponding ring of sets;
5. Measurable functions;Convergence almost everywhere;
6. Measurable functions and uniform convergence; the Egorov theorem;
7. Definition and properties of measurable functions;
8. Lebesgue integral;The definition and the properties of the Lebesgue integral;The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;
9. Comparison of the Lebesgue integral and of the Riemann integral.
10. Absolute continuity and σ-additivity of the Lebesgue integral; the Chebyshev inequality;
11. Lebesgue integral for simple functions;

Syllabus

• Lebesgue extension of a measure;

Extension a measure from a semi-ring of sets to the corresponding ring of sets;

Algebras and σ-algebras of sets;

Measure spaces;

Lebesgue extension of a measure defined on a ring of sets;

Properties of the Lebesgue measure;

• Measurable functions;

Definition and properties of measurable functions;

Convergence almost everywhere;

Measurable functions and uniform convergence; the Egorov theorem;

• Lebesgue integral;

Lebesgue integral for simple functions;

The definition and the properties of the Lebesgue integral;

Absolute continuity and σ-additivity of the Lebesgue integral; the Chebyshev inequality;

The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;

Comparison of the Lebesgue integral and of the Riemann integral.

### Contact Teaching Time

Information on contact teaching time is available from the course guide.

### Teaching Breakdown

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

### Summative Assessments

#### Exam

Assessment Type Weighting Summative 80 Students will be invited to contact Course Coordinators for feedback on the final examination.
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

#### Homework

Assessment Type Weighting Summative 10 In-course assignments will normally be marked within one week and feedback provided to students in tutorials.
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

#### Homework

Assessment Type Weighting Summative 10
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

### Formative Assessment

There are no assessments for this course.

### Resit Assessments

#### Best of written exam (100%) or written exam (80%) with carried forward in-course assessment (20%)

Assessment Type Summative
##### Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

### Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

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