What is the motivation behind the definition of Lebesgue measurable set?

Question

We study the definition of Lebesgue measurable set to be the following:

Let $A\subset \mathbb R$ be called Lebesgue measurable if $\exists$ a Borel set $B\subset A$ such that $|A-B|=0$,where $|.|$ denotes the Lebesgue outer measure of a set.

Then we have theorems like:

$A\subset \mathbb R$ is Lebesgue measurable iff

$(1)$ Given any $\epsilon>0$ there exists $F\subset A$ closed such that $|A-F|<\epsilon$.

$(2)$ Given any $\epsilon>0$ there exists $G\supset A$ open such that $|G-A|<\epsilon$.

I have two questions here.

First is that what motivates the definition of Lebesgue measurable sets and second is that why we are approximating Lebesgue measurable sets from below by closed sets and from above by open sets.I am studying the topic measure theory from Sheldon Axler's book that does not give motivation behind these definitions and theorems.Can someone give me motivation behind these things?

Answer 1

We certainly want to be able measure any bounded (open or closed) interval by its length. Also, in an attempt to find as many measurable sets as possible, we can certainly agree that the measure of finitely disjoint measurable sets should be the sum of the respective measures. One can readily take this further to countable unions of disjoint measurable sets by defining its measure as the supernumerary of the measures of finite sub-unions. Similarly for the other operations used in the definition of Borel set. The final step is to extend this to Lebesgue sets by postulating that a set that hardly differs from a Borel set (namely differs only by a set that should clearly contribute nothing) shall have the same measure as said Borel set.

This gives us a large family of sets where it is quite clear how we must define their measure (and as it turns out, can define that way consistently). On the other hand, for any set that is not obtainable in this way, we have no idea how to consistantly assign a measure to it.

Answer 2

All these allude to general properties in measure theory.

• Lebesgue measurability relates to a complete measure space. i.e., being able to talk about any set that should have measure zero.
• Theorem (1) relates to the notion of inner regular measure. This means that knowing measures of contained compact sets tells you the measure of the set. Similar, in my mind, to how one can tell the value of a continuous function at a point by a sequence of values.
• Theorem (2) relates to the notion of outer regular measure. This means you can know the measure of a set by measures of containing open sets.

Lebesgue measurable sets as well as Borel sets can be pretty crazy, which is why you would prefer to have some intermediate better known sets to help you check the measure.

As for why are people talking about Lebesgue measurable sets, instead of just Borel sets, I think it comes from the fact that sets with measure $0$ "might as well be the empty set", qualitatively. But you want to ensure that you can discuss all sets which should have measure zero. That's why you look at sigma algebra which adds them artificially.

That's my intuition anyway. Perhaps someone can give you a better explanation.