continuity of probability

Continuity of probability says that probabilities respect limits of growing or shrinking events. If you have a chain of events that keeps expanding — A1 inside A2 inside A3 and so on, each containing the last — then the probability of their eventual union (the limit you grow toward) is the limit of the individual probabilities: P(the union) = limit of P(An). The mirror statement holds for events that keep shrinking, nesting down toward their intersection. In short, you can pass the limit inside or outside the P without harm.

Here is why it is true, in one breath: continuity is countable additivity wearing different clothes. Given a growing chain, slice it into the disjoint new pieces gained at each step (A1, then A2 minus A1, then A3 minus A2, and so on). Countable additivity adds these infinitely many disjoint slivers to give the probability of the whole union, and the partial sums of that series are exactly the P(An) climbing toward the answer. So 'probability is continuous along monotone sequences' and 'probability is countably additive' are two faces of the same axiom.

Continuity is the bridge from finite reasoning to genuinely infinite questions, and it is what lets you take limits with confidence. It justifies statements like 'the probability that you eventually succeed equals the limit of the probability that you have succeeded by trial n', and it underlies tail events, the Borel-Cantelli lemmas, and convergence theorems further on. It is quietly load-bearing: without it, infinite probability arguments would have no licence to swap limits and probabilities.