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**cfp** There are a continuum of independent uniform random variables indexed by $\displaystyle i\in[0,1]$. For each such $\displaystyle i$, $\displaystyle U(i)$ is the realisation of that random variable, that is to say, a sample from it. (So a value in $\displaystyle [0,1]$.)

(If you want to be precise the state space $\displaystyle \Omega$ is the set of all functions from $\displaystyle [0,1]$ to $\displaystyle [0,1]$, and each random variable is a functional $\displaystyle p_i$ on this space defined by $\displaystyle p_i(\omega)=\omega(i)$. If $\displaystyle \omega\in\Omega$ is the realised state then $\displaystyle U(i)=p_i(\omega)=\omega(i)$.)

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.