TITLE: What is a realization of random variable?
QUESTION [8 upvotes]: Given a probability space $(\Omega, \mathcal{F}, P)$, we define a real-valued random variable $X$, which is a measurable function from $(\Omega, \mathcal{F})$ to $(R,\mathcal{B}(R))$. 
Since $X$ is a function, what's the precise definition of one realization of $X$? 
When I use Matlab to simulate one realization of $X$, I am actually using pseudo-random sequences, which are deterministic but statistically very close to the real distribution of $X$.
Is the realization of $X$ only an approximative definition? Can we ever really have one true realization of $X$?

REPLY [3 votes]: Given your set-up, a realisation of $X$ is no more than $X(\omega)$ for some $\omega\in \Omega$.