I am planning to teach a course on measure theory in MSc. first year.I always like to give motivation behind what I am teaching.I am taking the following as the definition of Lebesgue measurable set.
$A\subset \mathbb R$ is said to be a Lebesgue measurable set if for each $E\subset \mathbb R$ we have, $\mu^*(E)=\mu^*(E\cap A)+\mu^*(E\cap A^c)$.
Now I want to derive the equivalent characterizations that $A\subset \mathbb R$ is Lebesgue measurable $\iff$ for each $\epsilon>0$ there exists an open set $G_\epsilon\supset A$ such that $\mu^*(G_\epsilon\setminus A)<\epsilon$.$\iff$ for each $\epsilon>0$ there exists a closed set $F_\epsilon\subset A$ such that $\mu^*(A\setminus F_\epsilon)<\epsilon$.
Now I am trying to give some motivation behind these two characterizations.But I am finding nothing that I can tell about these things.I am eager to know why Lebesgue thought of approximating measurable sets from outside by open sets and from inside by closed sets.Can someone provide me some motivation?
Best Answer
Topologically, these are the most simple sets. If you want to try to approximate measurable sets by simple sets, that is a natural thing to do. Also, since finite sets are meant to have Lebesgue measure zero and nonempty open sets are meant to have positive Lebesgue measure, that is the natural approximation.
It is also generally true that for any (completion of a) $\sigma$-finite Borel measure, one can approximate measurable sets from the inside by closed sets and from the outside by open sets. Teaching this result might provide another path.
It is worth noting that the restriction of Lebesgue measure to the Borel sets is already due to Borel and Lebesgue relied on the work of Borel and others, like, Jordan. For references to the history of Lebesgue's theory, see here.