MX4083: MEASURE THEORY (2020-2021)

Last modified: 05 Aug 2021 13:04

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
Session	First Sub Session	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?

MA1511 Set Theory (Passed)
MA2009 Analysis I (Passed)
Any Undergraduate Programme (Studied)
MA2509 Analysis II (Passed)

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?

Course Description

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

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;

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

More Information about Week Numbers

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

4 assignments (25% each)

Alternative Resit Arrangements for students taking course in Academic Year 2020/21

Resubmission of failed elements (pass marks carried forward).

Formative Assessment

There are no assessments for this course.

Course Learning Outcomes

Knowledge Level	Thinking Skill	Outcome

Factual	Remember	ILO’s for this course are available in the course guide.