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But if we measure the outcome of an exact value, we get a line segment, which of course has an area of 0. But that value is in no way impossible.

Then I remembered that the sum of all events' probability is 1. So with an uncountably infinite amount of outcomes in a sample space, they must all sum to 1, yet when looking at one individually its probability is 0.

It kind of paralleled integral calculus to me in that the probability of an exact value is analogous to a differential element of probability. Can anyone shed some light on this idea?