This week is about Vitali covering Lemma and its applications, Lebesgue density theorem.

Pugh: Ch6: Ex 39, 48, 53, 58, 66

1. Tao Ex 8.3.2,
2. Tao Ex 8.3.3
3. Now we have covered the main results in Lebesgue measure theory, can you try to summarize the key steps and how to prove that? You can share it on your website. You can follow either Tao or Pugh's approach.
4. Read Pugh 6.8.

Tao-II: 8.2.7, 8.2.9, 8.2.10

Next week, we are going to consider Fubini's theorem in Tao 8.5 (see also Pugh section 6.7). You can read ahead. You may want to review the fact that, if a non-negative series is convergent, then any rearrangement of the series is convergent (the partial sum will form a monotone bounded sequence), hence a 'double series' of non-negative terms $\sum_n \sum_m a_{nm} = \sum_m \sum_n a_{nm} = \sum_{N=0}^\infty \sum_{n+m=N} a_{nm}$.

In this week, we discussed Lebesgue integral, using the intuitive picture of undergraph.

0. write a short summary about Lebesgue integral, e.g how we define it, how does it compare with Riemann integrals. Share it on your homepage.

1. read Tao Analysis-II, 7.5 and 8.1.

2. Pugh Ex 25, 28

1. Read Pugh section 6.2, 6.4, 6.5. Read the proof of Theorem 21 and 26.

2. Pugh Ch 6,

1. Ex 3. Consider the special case of a diagonal line $\{y=x\}$ in $\R^2$. Can you prove directly that it has measure zero, i.e, using covering by boxes? You then may want to read Section 6.3 Theorem 9. Or, can you use today's zero slice theorem to prove it?
2. Ex 6. How to reduce the unbounded case to bounded case?
3. Ex 12. You can assume properties in Ex 11.