The Essay Problem - Mathematics 3MI - spring 2003

The written exam consists of two parts. The first 90 minutes are without any means of help ("closed book") and during this time an Essay Problem is do be finished and handed in. Here, one typically is asked to state and prove one or several specific propositions and theorems in a central topic from the course. In the remaining 90 minutes more traditional problems are to be solved, and here all usual means of help are allowed ("open book").

The essay problem will be one of the following 14 problems. Notice that a Danish translation is appended.

The (long) list with 14 topics is:

  1. Sigma-algebras and measurable maps (Sætning 1.2, The Borel sigma-algebraen, limits of measurable functions, rules for measurable functions).
  2. Measure (derivation of properties held by a measure, examples, almost everywhere).
  3. The integral of positive functions (in particular, Lebesgue's monotone convergence theorem).
  4. The integral of real functions (in particular, Fatou's Lemma and Lebesgue's dominated convergence theorem).
  5. The integral with a real parameter (Sætning 4.26 og 4.28).
  6. Topic 1 from the Fourier Transform
  7. Topic 2 from the Fourier Transform
  8. Uniqueness of the Lebesgue maeasure.
  9. Locally integrable functions and the first main theorem of calculus.
  10. Product measures (Sætning 6.6, Lemma 6.7, and a description of the product sigma algebra).
  11. Tonelli's and Fubini's Theorems
  12. Hölder's and Minkowski's inequalities.
  13. Fischer's completeness theorem.
  14. Density of Cc(Rk) in Lp. (Sætning 7.28 and page 7.19).

Den (lange) liste med 14 spørgsmål er:

  1. Sigma-algebraer og målelige afbildninger (Sætning 1.2, Borel-sigma-algebraen, grænseovergange af målelige funktioner, regneregler).
  2. Mål (udledning af egenskaber ved mål, eksempler, næsten overalt).
  3. Integral af positive funktioner (især Lebesgue's sætning om monoton konvergens).
  4. Integral af reelle funktioner (især Fatou's Lemma og Lebesgue's sætning om majoriseret konvergens).
  5. Integral med reel parameter (Sætning 4.26 og 4.28).
  6. Caratheodory's Sætning.
  7. Eksistens af Lebesguemålet.
  8. Entydighed af Lebesguemålet.
  9. Lokalt integrable funktioner og infinitisimalregningens hovedsætning.
  10. Produktmål (Sætning 6.6, Lemma 6.7 og omtale af produkt-sigma algebraen).
  11. Tonellis og Fubinis sætninger.
  12. Hölders og Minkowskis uligheder.
  13. Fischer's fuldstændighedssætning.
  14. Tæthed af Cc(Rk) i Lp. (Sætning 7.28 og side 7.19).
Hans Plesner Jakobsen