There are a continuum of independent uniform random variables indexed by

. For each such

,

is the realisation of that random variable, that is to say, a sample from it. (So a value in

.)

(If you want to be precise the state space

is the set of all functions from

to

, and each random variable is a functional

on this space defined by

. If

is the realised state then

.)

I perhaps should have been clearer in my original message. It's traditional in the field I work in to blur the distinction between random variables and samples from them, for notational convenience.