Let's say we conduct a random experiment. The possible outcomes may be too complicated to describe, so we instead take some measurement (in terms of real numbers) from it which are of interest to us. Mathematically we have defined a random variable X from the sample space/measurable space to where is the Borel sigma field. We then choose an appropriate probability for each Borel set, via a distribution function. Then, how does that distribution function determine the probability space in theory or proves its existence? What result is implicitly used here?
I understand the probability space on is well defined, and that is what practically concerns us, but in theory we also have a probability space which pushes forward its measure to the Borel sets. Hence given the distribution on we are pulling back to via . But what theorem guarantees that such a space exists?
Edit for clarification:For example, let us take weather of a city as the original sample space and the recorded temperature as the random variable. Suppose the distribution of is decided upon, based on empirical data. That's fine and now we can answer questions like 'what is '. We may not care about the probability space on since all the probabilities we wish to know relate to , but how do we know for sure that there exists a probability space on in the first place, and further, how do we know that a probability space on exists which would push its probability to yield the distribution of ? Unless we know that there exists such a probability space in theory, we will not be able to talk about expectation of , so the knowledge that such a space exists is crucial.
*Further edit: Note that we may not have an explicit complete mathematical model of the weather ever, but that does not bother us. We do have the ability however, of taking measurements of different types and thereby getting distributions related to temperature, pressure, wind speed, historical records etc. So practically we can answer questions related to these measurements. This is so what happens in practice I think. What I want is some mathematical theorem which assures me that there exists a model of the weather space, even though it is hard/difficult/impossible to find, which pushes its probability on to these distributions.