Maybe not the most formal of meanings, but my favorite is a probabilistic one: given a random element, how likely it is that it satisfies a predicate? If some elements don't, but it's still satisfied with probability 1, that's pretty clearly almost always.

EDIT: yeah you guys are right, I wouldn't worry too much about the prior not being a proper distribution, but still - this doesn't seem related to the cardinality of sets in a simple way after all!