Why is the inner measure problematic?

Question

Currently the Lebesgue measure is defined by the outer measure $\lambda^*(A)$ by the criterion of Carathéodory: A set $A$ is Lebesgue measurable iff for every set $B$ we have $\lambda^*(B)=\lambda^*(B\cap A) + \lambda^*(B\cap A^C)$. However, before the criterion of Carathéodory the Lebesgue measure was defined by the outer and the inner measure $\lambda_*(A)$. For Lebesgue a set has a measure $\lambda(A)$ iff the inner and the outer measure is the same: $\lambda_*(A)=\lambda^*(A)$. Thus, we dropped the inner measure in the current definition of the Lebesgue measure.

Why is the inner measure currently not used in the definition of the measure? What are the problems which occur by using the inner measure?

My attempts so far: I read that the criterion of Carathéodory makes the extension theorem easier. Why is this the case. Is this the only reason?

Answer 1

No comment on why C's version is "better", but it's not hard to show the two are equivalent - I never felt I understood the C definition until I realized this.

Hints/outline: First we wave our hands and claim that with both versions of the definition, $A$ is measurable if and only if $A\cap[n,n+1)$ is measurable for every integer $n$. So assume $A\subset[0,1]$.

It's easy to see from the definition of inner measure that $$\lambda_*(A)=1-\lambda^*([0,1]\setminus A).$$So $\lambda_*(A)=\lambda^*(A)$ if and only if $$\lambda^*([0,1])=\lambda^*(A)+\lambda^*([0,1]\setminus A) =\lambda^*([0,1]\cap A)+\lambda^*([0,1]\setminus A).$$

Answer 2

Tao talks about this in his measure theory book, which last time I checked was still freely available.

He says roughly that there is an asymmetry here because when measuring finite unions of boxes, the measure is subadditive and not superadditive. This leads to the fact that the Jordan inner measure $$ m_{*, J}(U) := \sup \{m(A)\ :\ A \subset U\ \text{is a finite union of boxes}\} $$ doesn't care whether or not you put finite or countable in the definition (because you are taking a supremum and its subadditive). So a natural Lebesgue inner measure where you put countable here gives no increase in power/resolution of the measure

Thinking like this leads to the idea that Lebesgue inner measure should just be the outer measure of the complement.