Overview

The theory and applications of the Lebesgue integral via measure theory. Application of ideas and methods from real analysis can be used to solve a variety of problems in mathematics.

MATH423 introduces the theory of the Lebesgue integral via measure theory. The study of measure and integration shows how the application of ideas and methods from real analysis can be used to solve a variety of problems in mathematics.

Paper title Measure and Integration Mathematics 0.0833 10 points Semester 1 (On campus) \$620.00 Tuition Fees for international students are elsewhere on this website.
Restriction
MATH 401-412
Limited to
BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc
Contact

Mathematics 400-level programme coordinator: Dr Fabien Montiel fabien.montiel@otago.ac.nz

Teaching staff

Dr Tim Candy

Paper Structure

Main topics:

Fundamentals/Review

• Riemann integral and limits, the problem of area.
• Sets and cardinality, theorems of Schröder, Bernstein, and Cantor.
• Axiom of choice, Zorn’s lemma, and the well-ordering principle.

Set Algebras

• Pi, lambda, and sigma algebras.
• Dynkin’s lambda pi theorem.

Measures

• Semi-continuity of measure, the outer measure.
• Measurable sets.
• The Caratheodory extension theorem.
• The Lebesgue measure.

The Lebesgue Integral

• Measurable functions, simple functions.
• Constructing the Lebesgue integral, basic properties.
• Lebesgue vs Riemann integral.

Key Convergence Theorems

• Monotone convergence and Fatou’s Lemma.
• The dominated convergence theorem.
• L^p spaces and completeness.

Fubini’s Theorem

• Product algebras/measures, measurability of projections.
• Tonelli’s Theorem.
• Fubini’s Theorem.

Applications and Extensions

• Fourier analysis.
• Probability theory.
Teaching Arrangements

Lectures: 18 lectures (50 minutes each)

Tutorials: Weekly drop-in sessions for help with assignments

Assessment: Two written assignments (60%) and a final take home exam (40%)

Critical Thinking, Interdisciplinary Perspective, Lifelong Learning.

Learning Outcomes

On completion of the study of this paper, students are expected to:

• Understand standard objects and results in measure theory.
• Understand the construction of the Lebesgue integral, and the key convergence theorems.
• Have a basic knowledge of applications of measure and integration to other areas of mathematics (for instance probability theory, Fourier analysis, and functional analysis)
• Understand how ideas and methods from real analysis can be used to solve a wide variety of problems in mathematics.

Timetable

Semester 1

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Blackboard