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?