TMA4225 Foundations of analysis: Fall term 2014
Exercises
– now in reverse order, newest on top.
| Week | # | § | pp. | Exercises | Remarks |
|---|---|---|---|---|---|
| 47 | 12 | 8.4 | 299–301 | 48, 49, 52, 57, 58 | If these problems are too many, we might take some lecture time to look at them. |
| Exam August 2013 (2013-08-07): 2, 5, 6 | |||||
| 46 | 11 | 8.1 | 280–281 | 17*, 18 | *I don't see how to do 8.17c right now, so don't feel bad if you don't either, and don't waste too much time on it. I haven't given up yet myself. Update: I found a solution and presented it in class. |
| 8.3 | 291–292 | 33 (hint: rapid oscillations) | |||
| Extra exercise | |||||
| 5.5 | 176–177 | 93 | |||
| 45 | 10 | 6.3 | 210–212 | 42, 43, 45, 51, 53 | |
| 6.4 | 220–223 | 64 | |||
| One extra exercise | |||||
| 44 | 9 | Some exercises that are not in the book | |||
| 43 | 8 | 6.1 | 190–191 | 5, 6, 18–20 | |
| Assume that \((\Omega,\mathscr{A},\mu)\) is a probability space, and \(E_n\in\Omega\) for all \(n\in\mathbb{N}\). Assume that \[\sum_{n=1}^\infty\mu(E_n)<\infty.\] Let \(F\) be the set of points in \(\Omega\) that belong to \(E_n\) for infinitely many \(n\). (In the language of probability, “\(E_n\) happens infinitely often”.) Prove that \(\mu(F)=0\). This is known as the first Borel–Cantelli lemma. | |||||
| 42 | 7 | 5.3 | 164–165 | 58, 63 | |
| 5.4 | 172–173 | 67, 69, 74, 81b, 82 | |||
| Assume \(f_n\to f\) a.e., where \(f_n\) are measurable functions on some measure space with \(|f_n|\le g\), and \(g\) is integrable. Show that then \(f_n\to f\) in measure. Hint: The set \(\{x\colon g(x)\ge\varepsilon\}\) has finite measure. | |||||
| 41 | 6 | 5.1 | 149–151 | 5, 12, 13, 16, 17 | |
| 5.2 | 156–158 | 21, 22 | |||
| 40 | 5 | 4.2 | 127–129 | 23, 26 | Hint for 4.43: Adopt the proof of DCT. |
| 4.3 | 137–139 | 41, 43–46 | |||
| 4.4 | 142–143 | 56 | |||
| 39 | 4 | 4.1 | 119–121 | 1, 5a, 11, 12, 16, 17 | Here is a file containing three solutions to 4.17. (Updated once.) |
| 4.2 | 127–129 | 18, 19 | |||
| 38 | 3 | Assume \(B_1\subseteq B_2\subseteq\cdots\) are Lebesgue measurable sets. Show that \[\lambda\Bigl(\bigcup_{k\in\mathbb{N}}B_k\Bigr)=\lim_{k\to\infty}\lambda(B_k).\] | Hint: Consider \(A_k=B_k\setminus B_{k-1}\). | ||
| 3.4 | 107–109 | 33–35, 42–44 | Hint for 3.34: Consider \(E_1\setminus E_k\). | ||
| 37 | 2 | 1.4 | 26–27 | 43, 52 | This exercise class was cancelled due to illness. I will write up solutions when I feel able to do so. |
| Extra exercise not in the book | |||||
| | 88–89 | | These problem should not have been given yet. I am sorry. | ||
| 36 | 1 | 1.1 | 9–10 | 8 | |
| 2.5 | 66–67 | 71, 74 | |||
| 3.2 | 94 | 15, 17–19 | |||
| Extra exercises not in the book | |||||