So far, we have climbed the 'one face' of the mountain Lebesgue measure, let's climb the other face. ¹⁾ 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.

In the following, the measurability is defined using Def 2 above.

(stolen from Tao's grad measure theory book)

Let $A$ be any subset of $\R^n$, then $m^*(A) = \inf \{ m^*(U) \mid U \supset A, U \text{ is open } \}$

If $\{E_i\}$ is a countable collection of measurable set, then $\cup_i E_i$ is measurable.

Every closed subset $A \In \R^n$ is measurable.

Hint: This is a hard one. First prove that $A$ can be written as countable union of bounded closed subsets, then suffice to prove the claim that any bounded closed (hence compact) subset $A$ is measurable.

What does a closed set look like? Say, the cantor set in $[0,1]$?

If $E$ is measurable, then $E^c$ is measurable.

Hint: Try to write $E^c$ as a countable union of closed sets, union a set of measure zero, hence is countable.