Probabilities over infinite sets are not at all meaningless. If a set is countable, they have to privilege some objects (in the sense that not everything can have the same probability). If the set is uncountable (say the real numbers between 0 and 1) then there's no problem with having a very well-behaved probability distribution. (I'm skipping over some details. The fact that not every set is measurable means that one needs to be very careful when one talks about meaningfulness of probability).