Probabilistic properties that are invariant under a change of sample space. This may be a standard question, but I couldn't find it anywhere. I am having trouble understanding why in many results in probability we can omit the actual structure $(\Omega, \mathbb{F}, \mathbb{P})$ of a probability space; for instance, when we have a sequence of i.i.d  Bernoulli random variables $X_n$, is the fact that the $X_n$ converge almost surely independent of the choice of the probability space? In the sense that the $X_n$ can be defined in many probability spaces while still being Bernoulli. What about for r.v variables with other distributions? Is there a general criterion that allows this "probabilistic" way of thinking?
Any resource that is related to this would also be very helpful. 

Terry Tao discusses exactly this matter here: https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/
