## MATH4MI Measure and Integration

First Semester |

In school and in introductory courses in mathematics, integral usually means Riemann integral of a real-valued function on the real line. This fundamental concept reveals its beauty in the Fundamental Theorem of Calculus which relates integration and derivation. However, the Riemann integral has some shortfalls that make it inadequate for many purposes in modern analysis. One of them is a gap in the fundamental theorem of calculus: the class of Riemann integrable functions does not coincide with the class of all functions that have an antiderivative. Another drawback is that the interchange of point-wise limits of function sequences and their integrals is only possible under rather restrictive conditions.

In this paper we introduce a modern theory of integration via measure theory that overcomes these shortfalls. It goes back to the French mathematician Henri Léon Lebesgue (1875–1941) and is the gate to many exciting branches of mathematics, like, for instance, modern probability theory, functional analysis, or the theory of partial differential equations. Topics include sigma-algebras, uniqueness and existence of measures (Carateodory's theorem), measurable mappings, the construction of the Lebesgue integral, convergence theorems and basic function spaces.

### Period

2018, Semester 1.

### Prerequisites

None. MATH201 and MATH301 are strongly recommended.

### Lecturers

Petru A. Cioica-Licht (Room 212, phone 479 7783, email: pcioica@maths.otago.ac.nz)

### Lecture times

**Start:** Wednesday, February 28th, 9-11 am in Room MA229.

### Lecture place

TBA

### Further Reading

- R. L. Schilling,
*Measures, Integrals and Martingales*, Cambridge University Press, 2005. In course reserve. - D. L. Cohn,
*Measure Theory*, Second Edition, Birkhäuser, 2013. - N. V. Krylov,
*Introduction to the Theory of Dissufion Processes*, American Mathematical Society, 1995. - W. Rudin,
*Real and Complex Analysis*, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, 1974. - M. E. Munroe,
*Measure and Integration*, Addison-Wesley Publishing Company, Inc., 1953. - G. B. Folland,
*Real Analysis*, 2nd. Ed., Wiley, 1999.

### Internal Assessment

Short weekly assignments

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, 0.4E + 0.6A)**

where:

- E is the Exam mark
- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.