When you say "uniform random number in the set [0,1]", you mean something isomorphic to a countably infinite sequence of (independent) random bits (each equally likely 0 or 1) . The probability of any particular sequence is then 0. I guess you alluded to this with "probability (or rather density)".

But what's your larger point? Why is it necessary to fix some particular random bits? Couldn't you just as well say I have the ability to generate them if I want them? I guess by having them exist before I ask to see them, you can think of a Player decision node as existing frozen in time (interactions with the world happen only as results of decisions).