What are the advantages of outer measure?

Question

I am learning about measure theory. I have studied on outer measure then i am learning about Lebesgue measure. But i have a question why we learn outer measure since we have Lebesgue measure? That is What are the advantages of outer measure ?

Answer 1

The Lebesgue measure $\lambda$ is the measure induced(by Caratheodory extension theorem) from Lebesgue outer measure, which is defined as $$ \lambda^*(S) = \inf\left\{\sum_{j=0}^\infty (b_j - a_j)\mid S \subseteq \bigcup_{j=0}^\infty [a_j,b_j[\right\} $$ In the original papers of Lebesgue, he defined $\lambda$ by constructing it. But the common approach is to use a outer measure and induce $\lambda$ from that outer measure.

By Caratheodory extension theorem, the induced measure from an outer measure is automatically complete, and agrees with that outer measure on every measurable set.

Moreover, as a consequence of Caratheodory extension theorem, the action of $\lambda$ is uniquely determined by its action on intervals and union of intervals. Therefore, it is easy to show the translation invariancy of $\lambda$ if you define it this way.

So by making the construction of measures more abstract, it is actually simpler to do a lot of the proofs.

Answer 2

Another reason besides the excellent answer given is that every set has an outer measure, whereas not every set has a measure. If a set can't be proven to be measurable, it's common to investigate it with the outer measure. If the set turns out to be measurable, the outer measure results still apply because they agree