Countable additivity axiom

If we have an infinite sequence of disjoint events, Then the probability of the union of these events, of these infinitely many events, is the sum of their individual probabilities.

There are two kinds of infinite sets, countable and uncountable sets.

There are sets whose elements can be arranged in a sequence, like the integers.

The unit square or the real line are uncountable, because their elements cannot be arranged in a sequence.

Area is a legitimate probability law. It does indeed satisfy the countable additivity axiom as long as we only deal with nice subsets of the unit square.