Suppose that we have a set $\Omega\times [0,1]$ where $\Omega$ is finite and stands for the state space, and $X$ is a random variable that is indepedent of $\Omega$ and uniformly distributed on $[0,1]$ and denote as $x$ a realization of $X$.
What does it meand that $X$ is uniformly distributed on $[0,1]$? If for example $\Omega=\{a,b,c,d\}$ would this mean that each state of $\Omega$ is equaly likely to occur?