I'm reading Royden's real analysis 4th edition, and he defines a real set $E$ to be lebesgue measurable if, for all real sets $A$, $m(A)=m(A∩E)+m(A∩E^c)$. Here, $m$ is the outer measure of a set. I believe this is called Caratheodory's criterion.
Now, I'm also reading Baby Rudin (Principles of Analysis), and in chapter 11, Rudin defines a real set to be (lebesgue) measurable if it is the union of a countable collection of finitely measurable real sets, and he has another definition for "finitely measurable" which is a bit lengthy to type up.
I'm having trouble proving that these two definitions are equivalent; that is, I want to show that the set of lebesgue measurable sets constructed under Royden's definition is the same as the set of lebesgue measurable sets constructed under Rudin's definition. Can someone who is familiar with both definitions (especially rudin's definition) help me out by suggesting a proof or a text that may help?
So far, I managed to prove that a set measurable under rudin's definition satisfies the caratheodory criterion. I did this by using the following facts: $m(A)=m(A∩E)+m(A∩E^c)$ if and only if for every $\epsilon >0$, there exists an open set $O$ such that $m(O\setminus E) \le \epsilon$. Also: If a set $E$ is measurable under Rudin"s definition then there exists an open set $O$ such that $m(O\setminus E) \le \epsilon$. Can somebody help me out with the converse?