Dato | Undervises av | Sted | Tema | Kommentarer / 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 2.28Further 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 thatµ{x : f(x)=∞} = 0andµ{x : f(x)>0} is σ-finite (that is, a countable union of sets of finite measure). | ||
22.08.2011 | Terje Sund (TS) | B63 | Product measures. | Product sigma algebras. Existence and uniqueness of product measures. The theorems of Fubini and Tonelli. |
24.08.2011 | TS | B71 | Product measures. Exercises | Set 1: Exercise 1, 2, and 3. Folland 2.28 (a), (c) |
29.08.2011 | TS | B63 | Product measures. Transformations of measures. | |
31.08.2011 | TS | B71 | Transformations of measures. Exercises | Set 2: In Teschl: Problem 7.14, 7.16MAT 4300 exam, December 3, 2004, Problem 4 |
05.09.2011 | TS | B63 | Decomposition of measures. | The Lebesgue decomposition. The Radon-Nikodym theorem. |
07.09.2011 | TS | B71 | Decomposition of measures. Exercises | Set 3: In Teschl:Problems 7.18, 9.1, 9.2, 9.3 |
12.09.2011 | TS | B63 | Complex measures. | |
14.09.2011 | TS | B71 | Complex measures. L^p duality. Exercises | Set 4: In Teschl:Problems 9.4, 9.5, 9.9Folland 3.17 (Assume the given measures are sigma-finite.) |
19.09.2011 | TS | 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. |
21.09.2011 | TS | 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. |
26.09.2011 | TS | 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 |
28.09.2011 | TS | B71 | Open Mapping and Closed Graph. Exercises | Set 6: Folland Chap. 6, exercise (E) 1, 2; Chap. 2 E 20, Chap. 6. E 10. |
03.10.2011 | TS | B63 | Banach Spaces | The Hahn Banach Theorem |
05.10.2011 | TS | B71 | Consequences of Hahn-Banach. Exercises. | Reflexive spaces. Weak convergence.Set 7: Folland Ch. 7 E 2, Exam December 2004, 1 and 3 |
10.10.2011 | TS | B63 | Weak convergence | Teschl § 4.3, Folland §5.4. |
12.10.2011 | TS | B71 | Weak convergence. Exercises | Set 8: Teschl Problem 4.1, 4.2, 4.3. Folland E 5.35 |
17.10.2011 | TS | B63 | Derivatives of measures | Teschl § 9.2. Folland § 3.4. Wiener covering lemma. Lebesgue differentiation theorem. |
19.10.2011 | TS | B71 | Derivatives of measures. Exercises | Set 9: Teschl Problems 4.6, 4.7, 4.8. Folland E 5.22 |
24.10.2011 | TS | B63 | Derivatives of measures. Modes of convergence | Folland § 2.4 (not convergence in measure) |
26.10.2011 | TS | B71 | Exercises | Set 10:Teschl Problems 4.12, 4.13, 4.14 (Misprint in Problem 4.12: Should read s-lim l_n =l). |
31.10.2011 | TS | B63 | Nets | Folland § 4.3. Nets are often useful in functional analysis. |
02.11.2011 | TS | B71 | Exercises | Remaining problems from previous weeks. |
07.11.2011 | TS | B63 | Exerises | Set 11: Teschl Problem 4.5, Folland E 5.26 |
09.11.2011 | TS | B71 | Exercises | Set 12: Eksamen i MAT 4300, Mandag 4. desember 2006, problem 2 & 4 |
14.11.2011 | TS | B63 | Recapitulation | We look at some of the main theorems once more. |
16.11.2011 | TS | B71 | Exercises | MAT 4300 exam, December 3, 2004, Problem 3 |
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Published Aug. 7, 2011 7:01 PM
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