How did The North Face get its name? The name of the company is based on the north face of the Half Dome in Yosemite, California, to which attention was given on the generalization that the north face of a mountain in the northern hemisphere is regarded as the coldest, iciest and thus the most formidable to climb.

So far, we have climbed the 'one face' of the mountain Lebesgue measure, let's climb the other face. Here are some guides. Our goal is to start from the alternative definition of measurability (see Lecture 4, bottom),

Def 2 : A subset $E$ is measurable, if for any $\epsilon>0$, there exists an open set $U\supset E$, such that $m^*(U \RM E) < \epsilon$.

and derive the same set of properties for measurable sets. Note that, all the properties of outer-measure can still be used. I am going to be lazy, and steal the proposition we need to prove from Tao's graduate level measure theory book.