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)


What is this called and how is it resolved?

It's called "lack of uncountable additivity". The property that for disjoint sets $A_i$
$$
P\left[\bigcup_{i\in S} A_i\right] = \sum_{i\in S} P[A_i]
$$
(additivity) is required among the axioms of probability only when $S$ is a finite or countable family of sets.
Why that? Because we only need countable additivity to prove all results that you know in probability. Requiring uncountable additivity would give you a more limited theory, since you cannot include in your theory continuous probability measures: it would not be possible to treat models such as the Lebesgue measure on $[0,1]$ and Gaussian distributions. These are arguably only abstract models, but they have a role in approximating real-world experiments.
So it's not a bug, it's a feature that allows for a richer probability theory and more setups that satisfy the axioms of probability.
