MAT4410 - Autumn 2011

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10.08.2011    Exercises for repetition.  It is strongly recommended that you refresh parts of the basic theory of measure and integration. This may be found in

Teschl Ch.7 and 8 (MAT 4400/3400)

or in Folland, Real Analysis, Ch. 1 and 2. We list below some useful exercises.

In Teschl Ch. 7: Problem 2, 5, 10, 11, 13, 14.

In Folland: Exercise 2.28

Further exercises:

Exercise 1.

(a) Let f be a nonnegative function on a measure space (X,M,µ). How is the Lebesgue integral of f defined?

(b) Let X= [a,b], M = the Borel σ-algebra on [a,b], µ= the Lebesgue measure on M. Explain that for step-functions s on [a,b] the Lebesgue integral is equal to the Riemann integral. Also explain that the Lebesgue integral of a continuous, non negative function f is equal to its Riemann integral.

Exercise2. Derive the Monotone Convergence Theorem from Fatou's Lemma.

Exercise 3. Let f be a nonnegative, integrable function on a measure space (X,M,µ). Show that

µ{x : f(x)=∞} = 0

and

µ{x : f(x)>0} is σ-finite (that is, a countable union of sets of finite measure). 

22.08.2011Terje Sund (TS)  B63  Product measures.   Product sigma algebras. Existence and uniqueness of product measures. The theorems of Fubini and Tonelli. 
24.08.2011TS  B71  Product measures. Exercises  Set 1: Exercise 1, 2, and 3. Folland 2.28 (a), (c) 
29.08.2011TS  B63  Product measures. Transformations of measures.   
31.08.2011TS  B71  Transformations of measures. Exercises 

Set 2: In Teschl: Problem 7.14, 7.16

MAT 4300 exam, December 3, 2004, Problem 4 

05.09.2011TS  B63  Decomposition of measures.   The Lebesgue decomposition. The Radon-Nikodym theorem.  
07.09.2011TS  B71  Decomposition of measures. Exercises  Set 3: In Teschl:

Problems 7.18, 9.1, 9.2, 9.3 

12.09.2011TS  B63  Complex measures.    
14.09.2011TS  B71  Complex measures. L^p duality. Exercises  Set 4: In Teschl:

Problems 9.4, 9.5, 9.9

Folland 3.17 (Assume the given measures are sigma-finite.) 

19.09.2011TS  B63  L^p duality. Riesz' Representation theorem.  Note that the proof of (6.14) in Folland is wrong (the functions f_n will usually have an infinite range, hence do not belong to S). We recommend the (simpler) approach given in Teschl Theorem 10.2. 
21.09.2011TS  B71  Riesz' Representation theorem. Exercises  Thm. 10.5 in Teschl; in Folland (7.2).

Set 5: In Teschl:

Problems 9.10, 9.11, 9.12, 9.13. 

26.09.2011TS  B63  (Riesz' Representation theorem.) Banach Spaces   Chap. 4 in Teschl. In Folland Chap. 5 §§ 1-3. Baire's Theorem. The Principle of Uniform Boundedness, the theorems of Open Mapping and Closed Graph 
28.09.2011TS  B71  Open Mapping and Closed Graph. Exercises  Set 6: Folland Chap. 6, exercise (E) 1, 2; Chap. 2 E 20, Chap. 6. E 10. 
03.10.2011TS  B63  Banach Spaces  The Hahn Banach Theorem 
05.10.2011TS  B71  Consequences of Hahn-Banach. Exercises.   Reflexive spaces. Weak convergence.Set 7: Folland Ch. 7 E 2, Exam December 2004, 1 and 3 
10.10.2011TS  B63  Weak convergence  Teschl § 4.3, Folland §5.4.  
12.10.2011TS  B71  Weak convergence. Exercises  Set 8: Teschl Problem 4.1, 4.2, 4.3. Folland E 5.35 
17.10.2011TS  B63  Derivatives of measures  Teschl § 9.2. Folland § 3.4. Wiener covering lemma. Lebesgue differentiation theorem. 
19.10.2011TS  B71  Derivatives of measures. Exercises  Set 9: Teschl Problems 4.6, 4.7, 4.8. Folland E 5.22 
24.10.2011TS  B63  Derivatives of measures. Modes of convergence  Folland § 2.4 (not convergence in measure) 
26.10.2011TS  B71  Exercises  Set 10:Teschl Problems 4.12, 4.13, 4.14 (Misprint in Problem 4.12: Should read s-lim l_n =l). 
31.10.2011TS  B63  Nets  Folland § 4.3. Nets are often useful in functional analysis. 
02.11.2011TS  B71  Exercises  Remaining problems from previous weeks. 
07.11.2011TS  B63  Exerises  Set 11: Teschl Problem 4.5, Folland E 5.26 
09.11.2011TS  B71  Exercises  Set 12: Eksamen i MAT 4300, Mandag 4. desember 2006, problem 2 & 4 
14.11.2011TS  B63  Recapitulation  We look at some of the main theorems once more. 
16.11.2011TS  B71  Exercises  MAT 4300 exam, December 3, 2004, Problem 3 
Published Aug. 7, 2011 7:01 PM - Last modified Feb. 7, 2020 4:08 PM
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