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    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.

    About this paper

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

    Mathematics 400-level programme coordinator: Dr Fabien Montiel

    Teaching staff

    Dr Tim Candy

    Paper Structure

    Main topics:


    • 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.


    • 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.
    • The Radon-Nikodym Theorem.
    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%)

    Graduate Attributes Emphasised

    Critical Thinking, Interdisciplinary Perspective, Lifelong Learning.
    View more information about Otago's graduate attributes

    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.


    Semester 1

    Teaching method
    This paper is taught On Campus
    Learning management system
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