My professor repeatedly emphasizes the importance of Caratheodory's theorem about characterization of measurability, but I don't get why it's so important. As far as I remember, I have never used this theorem when solving PBSets or writing proofs in this semester.
The theorem states
E is measurable if and only if for every set $A$, $|A|_e = |A \cap > E|_e + |A-E|_e$.
The professor said this theorem will be important in understanding abstract measure in the next semester. (But we only covered Lebegsue measure and Lebesgue integral this semester) Can someone explain about the importance or usage of this theorem in analysis? Does it become crucial in more abstract settings other than Lebesgue measure? It constantly bothers me because it feels like I am missing something important. I appreciate in advance.