MAT4410 - Autumn 2011

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DatoUndervises avStedTemaKommentarer / ressurser
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,?). How is the Lebesgue integral of f defined?

(b) Let X= [a,b], M = the Borel σ-algebra on [a,b], ?= 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,?). Show that

?{x : f(x)=∞} = 0

and

?{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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