Last time we had the definition of outer measure, and we basically followed Tao-II's presentation. This time, we will go through Lemma 7.2.5 (relatively easy) and Lemma 7.2.6 (about outer measure of a box, a bit hard). Pugh gives a different proof for the outer measure of a box being what it supposed to be, namely the naive volume, and he uses Lebesgue number. I am going to follow Tao's approach, although it is longer.
Then, we plan to talk about the construction of 'non-measurable set', in Tao-II, 7.3. And then, give the definition of measurable set, that follows the Caratheodory condition. There is an alternative and equivalent definition, see Tao-M (Tao measure theory grad textbook), which says $E \In \R^n$ is measurable, if for any $\epsilon>0$, there exists an open set $U \supset E$, such that $m^*(U\RM E) < \epsilon$, namely, measurable set are those than can be approximately from the outside by an open set.
We will use the discussion time, hopefully 30 minutes, to tackle Lemma 7.4.2, Lemma 7.4.4.