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 ?

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.
