Very little. "Almost all" in what sense? Cofinite? Cocountable? We can fairly safely assume they at least mean the subset satisfying P has Lebesgue measure 1, which means that we can define a countable collection of intervals around the exceptional set (not satisfying P) of arbitrarily small total length that if we know x is not in any of those intervals, then it definitely has property P. But that's not exactly intuitive, now, is it?
