## MA213-5-SP-CO:Riemann Integration and Lebesgue Measure

The details
2023/24
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Current
Monday 15 January 2024
Friday 22 March 2024
15
04 January 2024

Requisites for this module
(none)
(none)
MA203
(none)

(none)

## Key module for

BSC 5B43 Statistics (Including Year Abroad),
BSC 9K12 Statistics,
BSC 9K13 Statistics (Including Placement Year),
BSC 9K18 Statistics (Including Foundation Year)

## Module description

This module covers the Riemann Integral and its basic properties, integrability of continuous functions, and the Fundamental Theorem of Calculus as well as Improper Integrals.

The module will then focus on sequences and series of functions, and studies uniform convergence of functions and properties preserved under uniform convergence. Finally, the notion of Lebesgue integral is briefly introduced.

## Module aims

The aims of this module are:

• To introduce Rieman integration and show that every continuous function can be integrated. The Fundamental Theorem of Calculus will be proven.

• To develop methods for understanding when a function, defined as the limit of a sequence of other functions, is continuous, differentiable, integrable, and for differentiating and integrating this limit.

• To introduce the theory of Lebesgue integration, which extends the notion of the Riemann integral.

## Module learning outcomes

By the end of this module, students will be expected to be able to:

1. Develop a good working knowledge of the construction of the Riemann integral.

2. Understand the fundamental properties of the integral and the Fundamental Theorem of Calculus and its applications.

3. Understand uniform and pointwise convergence of functions together with properties of the limit function.

4. Understand the fundamental properties of Lebesgue integral.

## Module information

Indicative Syllabus

• Riemann integrals: Properties of the Riemann Integral, Fundamental Theorem of Calculus.

• Improper Integrals: Improper integrals and Differentiation of integrals and integrals depending on parameter.

• Sequences and Series of Functions: uniform and pointwise convergence of functions, continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions.

• The Lebesgue Theory: Basics properties of Lebesgue Measure and Integration.

## Learning and teaching methods

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography

The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.
Further reading can be obtained from this module's reading list.

## Assessment items, weightings and deadlines

Coursework / exam Description Deadline Coursework weighting
Coursework   Test
Exam  Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
Exam  Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period)

### Exam format definitions

• Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
• In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
• In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
• In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

Coursework Exam
20% 80%

### Reassessment

Coursework Exam
20% 80%
Module supervisor and teaching staff
Dr Jianya Lu, email: jianya.lu@essex.ac.uk.
Dr Jianya Lu
maths@essex.ac.uk

Availability
Yes
No
No

## External examiner

Prof Stephen Langdon
Brunel University London
Professor
Resources
Available via Moodle
Of 20 hours, 19 (95%) hours available to students:
1 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.

Further information

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