If by Boole's inequality you mean the thing that I know as the union bound, then it is far from obvious to me that it is exactly the same as Bell's theorem. In

If by Boole's inequality you mean the thing that I know as the union bound, then it is far from obvious to me that it is exactly the same as Bell's theorem. In fact they seem to me to be completely different. Boole's inequality also does not tell you "you're doing the math wrong when defining your measure", it tells you that the probability that either A or B happens is not greater than the sum of the probability of A happening and the probability of B happening. I'm not sure what you mean in the first part of your comment, could you indicate specifically what probability measure is defined incorrectly (or not defined) in (e.g.) this paper? https://arxiv.org/abs/1508.05949 https://arxiv.org/abs/1508.05949