Before the criterion of Caratheodory a set $A$ was defined to be measurable iff the inner and outer measure of it coincide $\lambda_*(A)=\lambda^*(A)$. Caratheodory found a definition for measurable sets which only uses the outer measure: $A$ is measurable iff for each set $B$ we have $\lambda^*(B)=\lambda^*(B\cap A) + \lambda^*(B\cap A^C)$.
What is the intuition behind the criterion of Caratheodory? Why is the property $\forall B: \lambda^*(B)=\lambda^*(B\cap A) + \lambda^*(B\cap A^C)$ equivalent to $\lambda_*(A)=\lambda^*(A)$?
Background of my question: This is a follow up of an answer given by David C. Ullrich in my question Why is the inner measure problematic? With his answer I can provide an answer for my question for $A\subseteq B$ and $B$ being a set higher dimensional interval $[a,b)^n$ (or another set for which we already have define its measure). However I do not find a proof that $\lambda^*(B)=\lambda^*(B\cap A) + \lambda^*(B\cap A^C)$ leads to $\lambda_*(A)=\lambda^*(A)$ for $A\not\subseteq B$.
