Wieso macht Caratheodorys Definition messbarer Mengen Sinn?

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Zu beantwortende Fragen:

• [ ] Was ist die Intuition hinter Caratheordorys Definition?
• [x] Wieso ist diese Definition notwendig? -> https://math.stackexchange.com/questions/1400503/caratheodory-criterion-sufficient-but-is-it-necessary
• [ ] Wieso ist die Definition von Caratheodory äquivalent zu Lebesgues Definition messbarer Mengen -> https://math.stackexchange.com/questions/2806982/what-is-the-inutition-behind-the-caratheodory-criterion-for-measurable-sets

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Relevante MSE Threads:

• https://math.stackexchange.com/questions/2773414/are-the-caratheodory-measurable-sets-the-smallest-completion-of-borel-sets-in-ge
• https://math.stackexchange.com/questions/1186305/is-the-carath%c3%a9odory-definition-of-measurable-set-the-most-general

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Möglicherweise eine weitere wichtige Quelle: https://mathoverflow.net/questions/34007/demystifying-the-caratheodory-approach-to-measurability