Why is the probability of a set of outcomes possibly non-zero if each outcome's probability is zero in continuous probability? In continuous probability, $\text{Pr}(X=x) = 0$ for all $x$. However, $\text{Pr}(a \leqslant X \leqslant b) >= 0$ given $b > a$. The $>$ is key here: each individual outcome has zero probability but the probability that an outcome occurs in a SET of outcomes can be non-zero.
That the probability of each element of a set is zero but the cumulative probability of the set can exceed zero is quite paradoxical.  What is this called and how is it resolved? Is it simply a property of probability limits by assumption?
(Posting this here rather than MathExchange as it is a statistical theory question)

I can't really say much about the mathematics of infinitesimals and the probabilities of points in a continuous space, but a trivial answer is that even though the probability of a point in that continuous space is defined as being zero, as long as there is an integral of your set of points (i.e. the points are contiguous) that integral will be non zero.
Yes, that answer is probably not helpful, but this next bit might be. The question is only relevant in theory because in practice you cannot observe or record or compare any result with infinite precision. In practice the real-world analogue of a continuous distribution is unavoidably granular and the probability of each point in that granular distribution will be non-zero.
