The following is the definition of a random variable:
Let be a probability space. A random variable is a real-valued function X on such that for all
I don't understand how this makes sense if our choice of can be arbitrary.
Let's say we are rolling a biased dice such that each side has a different probability, and the information we know about the system is -- then for this , according to the definition, is not a random variable (because you could construct a (disjoint) Borel set which covers the probabilities of e.g. , and is not in ). How does this make sense?