We do statistics on data which we assume to be outcome of random trials. The question however, is -- are we assuming that the outcome can be represented as a random variable?
The outcome of an experiment is representable as a random variable if that function is measurable. If the mapping from the event space, to the set of real numbers, is a measurable function, then we can even think of assigning a probability to that random variable.
Thought experiments are fine. We can assume that the outcome of a thought experiment, like rolling a dice, has an outcome that can be represented by a measurable function which we designate as the random variable.
When dealing with real data, don't we assume that the process that generates that data, can be represented as a random variable? Is that simply an assumption, or are there concrete reasons/tests to believe that it is indeed measurable?