Property (c) is known as countable additivity. It is not something that can be proved, but it is like an assumption or an axiom. In our experience, we have seen that operating as if the assumption is correct leads to useful and credible answers to many problems, so we accept it as a reasonable assumption. Not all probabilists agree that countable additivity is natural.
—from Fundamentals of Probability: A First Course, a book by Anirban DasGupta
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