Exercises
First week (Week 34)
- lecture
Second week (Week 35)
- The following exercises in the textbook: Exercise 1.1.1 (Page 5), Exercise 1.1.3 (Page 8), Exercise 1.1.5 (Page 10), Exercise 1.1.18 (Page 13).
Third week (Week 36)
- The following exercises in the textbook: Exercise 1.1.19 (Page 13), Exercise 1.1.20 (Page 15), Exercise 1.1.23 (Page 16, you can use the result of Exercise 1.1.22 if you want), Exercise 1.1.25 (Page 17).
Fourth week (Week 37)
- The following exercises in the textbook: Exercise 1.2.1 (Page 18), Exercise 1.2.2 (Page 18), Exercise 1.2.3 (Page 21), Exercise 1.2.4 (Page 25).
Fifth week (Week 38)
- Prove Lemma 1.2.9 on page 28 in the textbook.
- Exercise 1.2.5 on page 29 in the textbook.
- Prove Lemma 1.2.12 on page 30 in the textbook.
- Exercise 1.2.6 on page 31 in the textbook.
Sixth week (Week 39)
- Prove Lemma 1.2.13 on pages 31, 32 in the textbook.
- Exercise 1.2.7 on page 34 in the textbook.
- Exercise 1.2.8 on page 34 in the textbook.
- Exercise 1.2.9 on pages 34, 35 in the textbook.
Seventh week (Week 40)
- Exercise 1.3.1 on pages 53, 54 in the textbook.
- Given a complex-valued simple function as in Definition 1.3.2 on page 50 in the book, show that its absolute value is an unsigned simple function.
Eighth week (Week 41)
- Prove Lemma 1.3.9 on pages 57, 58 in the textbook.
- Exercise 1.3.3 on page 60 in the textbook.
- Exercise 1.3.5 on page 61 in the textbook.
- Exercise 1.3.9 on page 63 in the textbook. You can use the results of Exercises 1.3.7 and 1.3.8 if you want.
Ninth week (Week 42)
- Exercise 1.3.10 on page 64 in the textbook. You can use the results of Exercise 1.2.11 if you want.
- Exercise 1.3.11 on page 65 in the textbook. You can use the results of Exercise 1.3.4 if you want.
- Exercise 1.3.21 on page 70 in the textbook.
- Exercise 1.3.22 on page 71 in the textbook.
Tenth week (Week 43)
- Prove (i) and (ii) of Theorem 1.3.20 on page 72 in the textbook. You can use the results of Exercise 1.2.16 if you want.
- Exercise 1.4.14 on page 87 in the textbook.
- Exercise 1.4.26 on page 94 in the textbook.
- Exercise 1.4.27 on page 94 in the textbook.
Eleventh week (Week 44)
- Consider the vertical truncation property of the unsigned integral (Property (x) of Exercise 1.4.36 on pages 100 and 101 in the textbook, note that in the book this was accidentally called "horizontal truncation"). What happens if the increasing sequence is replaced by a decreasing one (and, of course, the union is replaced by an intersection)? If you think that this remains true, give a detailed proof. If you think that this fails to be true in general, provide a counterexample and think about what additional assumptions one could impose to ensure that equality holds after all.
- Exercise 1.4.40 on pages 103 and 104 in the textbook.
- The complex-valued case of Exercise 1.4.42 on pages 105 and 106 in the textbook.
- Consider the previous exercise (Exercise 1.4.42). What happens if the finite measure assumption is dropped? What if we keep the finite measure assumption, but replace uniform convergence by pointwise convergence?
Twelfth week (Week 45)
- Prove Theorem 1.4.49 on page 111 in the textbook.
- Exercise 1.4.46 on page 112 in the textbook.
- Exercise 1.5.2 on pages 117 and 118 in the textbook.
- Prove Proposition 1.5.7 on page 120 in the textbook.
Thirteenth week (Week 46)
- Exercise 1.6.6 on page 137 in the textbook.
- Prove Lemma 1.6.17 on page 143 in the textbook.
- Prove Proposition 1.6.34 on page 165 in the textbook.
- Exercise 1.6.43 on page 169 in the textbook. You can use Theorem 1.6.25.
Fourteenth week (Week 47)