# TMA4225 - Foundations of Analysis

### Examination arrangement

Examination arrangement: Written examination

Evaluation form Weighting Duration Examination aids Grade deviation
Written examination 100/100 4 hours D

### Course content

The modern concept of integral was introduced on April 29, 1901, in a short article by Henri Lebesgue. That article opened a new chapter of analysis. The flaws of the Riemann Integral will be pointed out, and the Lebesgue integral will be introduced to remedy the situation. Key concepts include measure theory including sigma-algebras, measurable spaces, measurable functions, outer measures, construction of the Lebesgue measure, and product measures. The course also covers the classical convergence theorems, Fubini's theorem, functions of bounded variation, and the fundamental theorem of integral calculus.

### Learning outcome

1. Knowledge. The student masters basic concepts from measure theory, including sets of measure zero, measurable functions, the Lebesgue integral and Lebesgue spaces. The student has an overview of the central results of the theory of Lebesgue integration, including convergence theorems and Fubini's theorem. Moreover, the student is familiar with applications of measure theory to probability theory.

2. Skills. The student is able to perform operations using the Lebesgue integral and Lebesgue spaces. Moreover, the student is able to apply integration theory in one or several variables to formulate and solve problems in mathematics and technology, including problems involving discontinuous data.

### Learning methods and activities

Lectures and exercises. The lectures will be given in English if the course is attended by students who don't master a Scandinavian language. If the course is taught in English, the exam will be given only in English. Students are free to choose Norwegian or English for written assessments. A makeup exam may be given as an oral examination.

### Further on evaluation

see «Teaching methods and activities».

### Course materials

Will be announced at the start of the course.

### Credit reductions

Course code Reduction From To
SIF5052 7.5

# Timetable

Detailed timetable

# Examination

Examination arrangement: Written examination

Term Statuskode Evaluation form Weighting Examination aids Date Time Room *
Autumn ORD Written examination 100/100 D 2017-12-12 09:00
• * The location (room) for a written examination is published 3 days before examination date.
If more than one room is listed, you will find your room at Studentweb.