## MA4MTI-Measure Theory and Integration

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Autumn term module
Pre-requisites: MA2RA2 Real Analysis II or MA2RAX Real Analysis
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3MTI Measure Theory and Integration
Module version for: 2017/8

Module Convenor: Dr Karl-Mikael Perfekt

Summary module description:

This module provides an introduction to measure theory and integration.

Aims:

The course gives an introduction to the general measure theoretical approach to integration. A unified approach to the concepts of length, area, surface and integrals is given.

Assessable learning outcomes:

By the end of the course, students are expected to:

• have a deep understanding the concept of measure, measurability and integral for general measure spaces and be able to solve problems involving these;

• have a deep understanding of limiting processes with Lebesgue integrals and be able to handle these rigorously.

This module will be assessed to a greater depth than the excluded module MA3MTI.

Students should develop an understanding of the concept of area, surface and length and general measure theory. They should reflect on the difference, advantages and disadvantages of Riemann versus Lebesgue integration.

Outline content:

Measurable sets, measure, construction of measures, measure zero sets, transformation theorem, measurable functions, construction of integrals, properties which holds almost everywhere, and limit theorems.

Brief description of teaching and learning methods:

Lectures supported by lecture notes, non-assessed problem sheets with solutions and guided independent study.

Contact hours:
 Autumn Spring Summer Lectures 20 Guided independent study 80 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Oral assessment and presentation 100

Other information on summative assessment:

Formative assessment methods:

Oral feedback during exercises given during the lectures.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:

Requirements for a pass:

A mark of 50% overall.

Reassessment arrangements:

Oral examination/presentation.