onderwijs

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Measure Theory

I will teach the course from the book Measure, Integral and Probability, by Marek Capinski and Ekkehard Kopp, Second edition. This book contains a very elegant introduction to this very important and beautiful subject.

There will be a midterm exam on October 27 (12.00-14.45), and a final exam on December 23 (8.45-11.30). The midterm exam will count for half of your grade, but you can only pass if your average grade over the two exams is at least 6,0 and you have scored at least 5,0 for each of the two exams. There is only one chance for the midterm exam; there will be no second possibility. Of course, for the final exam there will be an extra possibility later. On December 23, there is also the possibility to do the whole exam. There is a second final exam in February 2011. Only the full exam can be done on this date.

I plan to treat the following material:

• Midterm exam: Section 2.1 – 2.5; Section 3.1-3.4.
• Final exam: All previous material; Section 4.1-4.6; Section 6.1-6.4; Section 7.1-7.2.

If you want to make an appointment with me, please do so via our secretary Maryke Titawano at maryke@few.vu.nl.

Material and exercises per week will also be published via the announcements in Blackboard and this webpage.

September 5/8:

Material: Sections 1.2 and 2.1.

Exercises: 2.1, 2.2, 2.3. Extra exercises:

• Let A be the subset of the unit interval consisting of all points without the number 7 in their decimal expansion. Show that A is a null set. Is A countable or uncountable?
• Let B be the subset of the unit interval consisting of all points which miss some “block” a(1)a(2)…a(k) in their decimal expansion. Is B a null set? Explain and be very careful with your arguments.
• Show that the function f defined at the bottom of page 9 is Riemann-integrable.

September 12/15

Material: Section 2.2.

Exercises: Prove Proposition 2.5 and Proposition 2.8. Exercise 2.4 and 2.5. Extra exercises:

• Show that in the definition of outer measure we can restrict ourselves to "special" intervals, for instance only open intervals, or only closed intervals. Be very precise with your arguments.
• Compute the outer measure of the set of irrational numbers in [0,1]. Be again very precise.

September 19/22

Material: Section 2.3, beginning of Section 2.4.

Exercises: Proposition 2.13, Proposition 2.15 (Is property (ii) really inherited? Be careful... And is it clear in (iii) that if a set is measurable, that any translate of this set is measurable?), Exercise 2.6, Proposition 2.16. Extra exercises:

• Let E(1), E(2), E(3), ... be measurable subsets of the real numbers. Define the set F as the set of all numbers which are an element of infinitely many of the E(i)'s. Let G be the set of numbers which are element of all but finitely many of the E(i)'s. Show that both F and G are measurable sets. Which of the two sets F and G is the largest?
• Show that the following collections are sigma-fields: (1) The collection of all subsets of the real numbers; (2) The collection of all countable sets and their complements; (3) The collection of unions of left-open and right-closed intervals with integer endpoints (including plus and minus infinity); (4) The collection consisting of two sets only, namely the empty set and the set containing all numbers.

September 26/29

Material: 2.4 - 2.5

Exercises: Exercise 2.7, Proposition 2.7, Exercise 2.8. Extra exercises:

• Find a sequence of open sets O(n) which all cover the set of rational points, and whose measure is at most 1/n.
• Finish the proof of Theorem 2.25.
• Is the condition in Theorem 2.19 (ii) that m(A1) is finite really necessary? What is really needed instead? 
• Give an example of two sigma fields whose union is not a sigma field.
• Consider the sigma field consisting of all subsets of the real line. Define n by n(A)=0 if A is countable and n(A)=infinite if A is not countable. Show that n is a measure.
• Desrcibe the sigma field generated by the singletons.
• Let A(1), A(2),…, A(n) be a partition of the real line. Describe the smallest sigma-field containing A(1), A(2),…, A(n).
• Let A(1), A(2),…, A(n) be a partition of the real line. Suppose I have a measure on this sigma-field. Give a precise description of the completion of the sigma field with respect to this measure. (This may – of course – depend on the measure; treat all possible cases.)

October 3/6

Material: 3.1 - 3.4 until page 63 (inclusive)

Exercises: Exercise 3.1-3.2; Proposition 3.8, Exercises 3.3, 3.4. Extra exercises:

• Finish the proof of Theorem 3.3, including all details.
• See here for the remaining exercises of this week.

October 10/13

Exercises: Exercise 3.5, Proposition 3.15, Exercises 3.6. Exercises 4.1, 4.2. Extra exercises:

• Suppose F is a sigma-field. Show that F contains the Borel sigma-field if and only if every continuous function is F-measurable.
• A function f is called upper semi-continuous at x if for all ε>0 there is a δ such that |x-y|< δ implies f(y) < f(x) + ε. Show that if f is everywhere upper semi-continuous, then it is measurable.
• The exam of October 2010.

November 7/10

Material: Finish Section 4.3, Section 4.4, beginning of Section 4.5

Exercises: Propositions 4.20, 4.23, 4.28, 4.29, 4.32.  Exercises 4.6 and 4.8. See here for an extra exercise.

November 14/17

Material: Section 4.5, Section 6.1 and Section 6.2.

Exercises: Exercise 4.9 and Proposition 6.3. Extra exercises:

• Check carefully the claims made in Section 6.1.
• Show that the Dirichlet function of Example 4.34 is a.e. continuous.
• Prove the claim on page 100, third paragraph of Example 4.35 ("More generally....")
• Exercises 1, 3 and 4 of the (full) exam of December 2009. (Click here.)
• Last year, someone asked the following question (at least this is how I interpreted the question...): In the proof of Theorem 4.33(i), it is mentioned that the conclusion is possibly incorrect for points which are endpoints of partition intervals, of which there are only countably many - a null set. But, since we are free in the way we choose our partition, any point can be such an endpoint, and this is clearly not a countable or null set. What is your answer to this problem?

November 21/24

Material: First part of Section 6.3.Extra exercises:

• In Theorem 6.5 the assumption that the measures are finite is needed. Let m be Lebesgue measure on the real line, and let v be counting measure (on the real line), that is, v(A) is equal to the number of points in A. Take A={(x,y); x=y} en show that the integrals in (6.3) are not the same.
• How would you give the general proof of Theorem 6.5?
• In the text, results are proved for finite measures. Show that the results follow for sigma-finite measures, by a limit argument.
• Give an example of a monotone class which is not a sigma-field.
• Consider measures m en n. Suppose that m(y; (x,y) in E) = m(y; (x,y) in F); in other words, assume that the “vertical” sections of E and F have the same measure. Show that the product measure (mxn)(E) and (mxn)(F) coincide. (This is called Cavalieri’s principle.)
• See the here for an extra exercise.

November 28/December 1

Material: Finish Section 6,3; Section 6.4.

Exercises: Exercise 6.1 and 6.2; Proposition 6.16. Extra exercises:

• Give all details of the proof of Fubini's theorem.
• See here for two more exercises.
• Show that two-dimensional Lebesgue measure (built from rectangles - see Section 6.1) is the same as the completion of the product of one-dimensional Lebesgue measures. See the last paragraph of Section 6.3 for the main line of the argument.
• Exercise 5 from the Februari 2011 exam (see here).
• Prove Theorem 6.5 for the case of finitely many rectangles (in class we did the case of two rectangles).

December 5 and 8

Material: Section 7.2 until Theorem 7.7 (inclusive)

Exercises: Exercise 7.1, Proposition 7.2, Exercise 7.3, 7.4, 7.5 and 7.6, Proposition 7.6, Proposition 7.8. Extra exercise: Exercise 6 of the february 2010 exam (see here).

December 12 and 15

Material: Finish Section 7.2.

Exercises: Proposition 7.9, Exercise 7.7, 7.8 and 7.9. Extra exercises:

• Extend the proof of Theorem 7.12 to the sigma-finite case.
• Prove that the Lebesgue decomposition is unique.
• Exercise 5 of the exam of december 2009
• Exercise 3 of the exam of february 2010
.