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  6. MX4083: Measure Theory

MX4083: MEASURE THEORY (2025-2026)

Last modified: 14 Nov 2025 14:16


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 Undergraduate Level 4
Term First Term Credit Points 15 credits (7.5 ECTS credits)
Campus Aberdeen Sustained Study No
Co-ordinators
  • Dr Alexey Sevastyanov

Qualification Prerequisites

  • Programme Level 4

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

Syllabus

  1. 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; σ-additive and σ-semi-additive measures;
    Measure spaces; Lebesgue extension of a measure defined on a ring of sets; Properties of the Lebesgue measure.
  2. Measurable functions; Definition and properties of measurable functions; Convergence almost everywhere; Measurable functions and uniform convergence; the Egorov theorem.
  3. 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

More Information about Week Numbers


Details, including assessments, may be subject to change until 31 August 2025 for 1st Term courses and 19 December 2025 for 2nd Term courses.

Summative Assessments

Exam

Assessment Type Summative Weighting 70
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback

Students will be invited to contact Course Coordinators for feedback on the final examination.
Learning Outcomes
Knowledge LevelThinking SkillOutcome
ConceptualAnalyseTo compare the Lebesgue integral and the Riemann integral.
ConceptualAnalyseTo define a σ-algebra, a measure and a measurable function, to check the definitions in examples and to prove simple results (seen and unseen) about these concepts.
ConceptualAnalyseTo derive properties of integrals.
ConceptualApplyTo be able to apply the Chebyshev inequality.
ConceptualApplyTo state, prove and use the Monotone Convergence Theorem, Fatou’s Lemma and the Dominated Convergence Theorem.
ConceptualUnderstandTo understand the Lebesgue extension of a measure defined on a ring of sets.
ConceptualUnderstandTo be familiar with absolute continuity and σ-additivity of the Lebesgue integral and of a measure.
ConceptualUnderstandTo state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses.
FactualApplyTo state and illustrate the definitions of the concepts introduced in the course.
ProceduralApplyTo integrate a simple measurable function, a general integrable function.
ProceduralApplyTo use the methods and results of the course to solve problems at levels similar to those seen in the course.
ProceduralUnderstandTo demonstrate knowledge and understanding of proof techniques used in the course.

Homework

Assessment Type Summative Weighting 15
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials.
Learning Outcomes
Knowledge LevelThinking SkillOutcome
ConceptualAnalyseTo compare the Lebesgue integral and the Riemann integral.
ConceptualApplyTo be able to apply the Chebyshev inequality.
ConceptualUnderstandTo be familiar with absolute continuity and σ-additivity of the Lebesgue integral and of a measure.
ConceptualUnderstandTo state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses.
FactualApplyTo state and illustrate the definitions of the concepts introduced in the course.
ProceduralApplyTo use the methods and results of the course to solve problems at levels similar to those seen in the course.
ProceduralUnderstandTo demonstrate knowledge and understanding of proof techniques used in the course.

Homework

Assessment Type Summative Weighting 15
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback
Learning Outcomes
Knowledge LevelThinking SkillOutcome
ConceptualAnalyseTo define a σ-algebra, a measure and a measurable function, to check the definitions in examples and to prove simple results (seen and unseen) about these concepts.
ConceptualAnalyseTo derive properties of integrals.
ConceptualApplyTo state, prove and use the Monotone Convergence Theorem, Fatou’s Lemma and the Dominated Convergence Theorem.
ConceptualUnderstandTo understand the Lebesgue extension of a measure defined on a ring of sets.
ProceduralApplyTo integrate a simple measurable function, a general integrable function.

Formative Assessment

There are no assessments for this course.

Resit Assessments

Exam

Assessment Type Summative Weighting 100
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback

Best of (resit exam mark) or (resit exam mark with carried forward CA marks)
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
FactualApplyTo state and illustrate the definitions of the concepts introduced in the course.
ProceduralUnderstandTo demonstrate knowledge and understanding of proof techniques used in the course.
ConceptualAnalyseTo compare the Lebesgue integral and the Riemann integral.
ConceptualApplyTo state, prove and use the Monotone Convergence Theorem, Fatou’s Lemma and the Dominated Convergence Theorem.
ProceduralApplyTo integrate a simple measurable function, a general integrable function.
ConceptualAnalyseTo define a σ-algebra, a measure and a measurable function, to check the definitions in examples and to prove simple results (seen and unseen) about these concepts.
ConceptualUnderstandTo understand the Lebesgue extension of a measure defined on a ring of sets.
ConceptualUnderstandTo state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses.
ConceptualApplyTo be able to apply the Chebyshev inequality.
ProceduralApplyTo use the methods and results of the course to solve problems at levels similar to those seen in the course.
ConceptualAnalyseTo derive properties of integrals.
ConceptualUnderstandTo be familiar with absolute continuity and σ-additivity of the Lebesgue integral and of a measure.
 

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