Examination

The examination consists of
  • a written examination (5 credits)
  • an oral examination (2.5 credits) . The oral examination may only be taken by those students who passed the written examination. 
You can get 30 points on the written examination and 10 on the oral. To pass the course you need minimum 15 on the written and 5 on the oral. If you fail the oral you do not need to redo the written examination (assuming you passed it). To get the grade VG or Pass with Distinction, you need above 30 points in total.

Study advice for the oral examination

Key topics to focus on:
Chapter 1.
  • Sigma algebras: Definition and Corollary 1.1.3, construction of the Borel sigma algebra.
  • Measures: Definition and Proposition 1.2.5
  • Outer measures and Lebesgue measure: Understand key definitions. Understand why outer measures are needed for construction of Lebesgue measure. Understand on a shallow level Theorem 1.3.6 and how it leads to the Lebesgue sigma algebra.
  • Revisit interesting examples such as the Cantor set in 1.4.6, and a non-measurable set 1.4.8. Understand briefly why we can not measure all possible sets, and hence the need for sigma-algebras. 
Chapter 2.
  • Understand how the integral is constructed. 
  • Proof of Monotone Convergence Theorem, Fatou's Lemma and the Dominated convergence theorem
  • Understand briefly how the Lebesgue integral connects with the Riemann one, and in particular when and why Riemann formulas can be used to evaluate Lebesgue integrals.
  • Triangle inequality for integrals, Proposition 2.4.7
Chapter 3
  • Know all results in 3.1 and be able to prove some of them.
  • Proof of Hölder and Minkovski
  • Explain the construction of L^p
  • Completeness of L^p (with proof), why is it a Banach space.
Chapter 5
  • Construction of product sigma algebras
  • Briefly understand the construction of product measures (the dive into Dynkin classes is for VG only), but everybody should think about why these are needed in the construction, i.e. understand the problem, even if not understanding the solution.
  • Properly state and prove Fubini's and Tonelli's theorems.
Past Examination Papers