MAT4300 - Autumn 2007

Undervisningsplan

DatoUndervises avStedTemaKommentarer / ressurser
04.12.2007? ? Final Exam? 2:30 p.m. (3 hours)?
28.11.2007? ? Dec. 04: 2,3, 4? Dec.04

Some additional problems:

Dec. 06 (Norsk)

Dec. 06 (English)

Exercise

Solutions to Dec.06?

22.11.2007? ? June 03: 2; 10 P; Problem: Let F(x)= arctan(x) if x is nonnegative, F(x)=-1 if x<0. Find the Lebesgue-decomposition of the Borel Stieltjes measure associated to F with respect to Lebesgue measure.? Solution to Problem

Comment on June 03, 2b): The case b=0, a nonzero (or a=0, b nonzero) is far from obvious. It is clear that (a,0)E has zero outer measure, being a subset of the x-axis in the plane. Since the Lebesgue product measure is complete, it follows that (a,0)E is measurable. However, this does not imply that (a,0)E is element of the (smaller) sigma algebra B x B. A far more sophisticated argument is needed.

June 03:1

June 03:2

June 03:2e), 3

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21.11.2007? ? 10. Product Measures? Cross sections. The Theorems of Tonelli and Fubini.?
15.11.2007? ? 8 C, E, K, Q; 9T; (7 B, H, I, Dec.05: 3)? ?
14.11.2007? ? 8.Decomposition of Measures. 10. Product Measures? The Lebesgue Decomposition. Riesz's Representation Theorem (without proof). The Product Measure Theorem. (Cross Sections.)?
08.11.2007? ? 9 U; 6 E, F , H, J, K, L, N, P, U (7 A, B, H, I)? Solution to 6P?
07.11.2007? ? 8.Decomposition of measures? The Jordan Decomposition. The Theorem of Radon-Nikodym. (The Lebesgue Decomposition. Riesz's Representation theorem.)?
01.11.2007? ? 9 J, K, L, M, (U; 6 E, F , H, J)? ?
31.10.2007? ? 6. L_p spaces. 7.Modes of Convergence. ? H?lder's inequality. Minkowsky's inequality. Completeness. Essentially bounded functions.?
25.10.2007? ? Exam June1999: 2a),b); 9 F, G, H, I, (J, K, L, M)? ?
24.10.2007? ? 9.Generation of Measures. 6.L_p Spaces? Hahn's Extension Theorem. Lebesgue measure on the real line. H?lder's inequality will be proved using convexity of the exponential function. ?
18.10.2007? ? 5 N, (S,) T. 9 B, C, E. Exam June 1999: 1, (2 a, b)? Exam June 1999, 1

Exam June 1999, 2?

17.10.2007? ? 9.Generation of Measures? (Hahn`s Extension Theorem.) The algebra F of h-intervals. The length function on F. ?
11.10.2007? ? Mid-term exams? No problem session?
10.10.2007? ? Mid-term exams? No lecture?
04.10.2007? ? Exam Dec. 72, 1; 5 O, P, (Q,) R, (S, T)? Exam Dec. 1972?
03.10.2007? ? 9. Generation of Measures? The sigma algebra of measurable sets. The Carathéodory Extension Theorem.

Quiz 6

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27.09.2007? ? Exercises 5 A, B, C, D, L, M. In addition: 9.Generation of Measures? Problem session the first hour. Theory the second hour: Measure on an algebra. Outer measure.?
26.09.2007? ? 5. Integrable Functions? Quiz 5?
20.09.2007? ? Exercises 3 M; 4 C, H, I, J, K, (L, M, N)? ?
19.09.2007? ? 4. The Integral? Quiz 4?
13.09.2007? ? Exercises 2 V; 3 C, J, K, L, M; 4 C, (H, I, K)? Solution to 2V

Strict inequality in 3J?

12.09.2007? ? 4. The Integral? Quiz 3??
06.09.2007? ? Exercises 2 I, K, M, N, O, P, Q, R; 3 A, B, (C, J, K)? ?
05.09.2007? ? 2.Measurable functions, 3. Measures? ?Quiz 2?
30.08.2007? ? Exercises 2A, B, C, D, E, F, H, (I, K, and M). You are encouraged to do as many exercises as possible before each problem session? Extra problem: Prove that the Cantor set C has mesure 0 (Hint: Use that the sum of the lengths of all intervals removed when constructing C, is equal to 1.)

Solution?

29.08.2007 --- ? ---? 2.Measurable functions? Quiz 1

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23.08.2007Terje Sund? B70? Chap.2 Measurable functions? We prepare the construction of the Lebesgue Integral by defining sigma algebras and measurable functions.?
22.08.2007Terje Sund? B70? Introduction? We start with some important properties and weaknesses of the Riemann-Integral (RI). Then we define the extended real number system. ?
Published Aug. 22, 2007 2:23 PM - Last modified Nov. 29, 2007 3:19 PM
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