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Why is Carathéodory’s Extension Theorem?

Caratheodory’s extension theorem and uniqueness of the extension are usually covered (in that order) in any undergraduate measure theory course. This is my understanding of the results, rephrased in a way that is more intuitive and goes over why structures like Dynkin systems are natural to think about (as opposed to the arbitrary definitions added to the course just for the sake of finishing the proof).