What are the advantages of outer measure? 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 ?

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
