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Third-Cycle Courses

Faculty of Engineering | Lund University

Details for the Course Syllabus for Course MATM19F valid from Autumn 2018

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General
Aim
  • For many mathematical investigations the notion of
    integrability in the sense of Riemann, which is used in the basic courses, is insufficient. Above all, it is difficult to guarantee
    that the limit of a sequence of Riemann integrable functions is an integrable function. The aim of the course is to acquaint the postgraduate student with the Lebesgue integral, and important theorems valid for it. This theory is indispensable for researchers in, e.g., mathematical analysis, numerical analysis or stochastic processes.
Contents
  • Basic theory of Lebesgue integration: basic measure theory, construction of the Lebesgue measure, convergence theorems and Fubini's theorem.
Knowledge and Understanding
  • For a passing grade the doctoral student must
  • be able to account for basic concepts and methods within theory of integration.
Competences and Skills
  • For a passing grade the doctoral student must
  • be able to use the theorems in the course to solve mathematical problems, in particular such that are relevant for applications such as if it is permitted to change the order of integration in iterated integrals, and if the integral of the limit of a given sequence of functions equals the limit of the integrals of the functions in the sequence,

    be able to formulate and prove the main theorems in the course.
Judgement and Approach
  • For a passing grade the doctoral student must
Types of Instruction
  • Lectures
  • Seminars
Examination Formats
  • Written exam
  • Oral exam
  • Written assignments
  • Failed, pass
Admission Requirements
  • At least 60 credits in mathematics as well as English B or the equivalent are required.
Assumed Prior Knowledge
  • Calculus in one and several variables. Linear algebra.
Selection Criteria
Literature
  • Cohn, Donald L.: Measure Theory: Second Edition. Birkhäuser, 2013. ISBN 9781461469551.
Further Information
Course code
  • MATM19F
Administrative Information
  • 2018-08-24
  • Professor Thomas Johansson

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