If you don’t have a given joint pobability space, you implicitly construct it (for example, by saying RV are independent, you implicitly construct a product space). Generally, the fact that sometimes you talk about X living on one space (on its own) and other time on the other (joint with some Y) doesn’t really matter, because in most situations, probability theory is specifically about the properties of random variables that are independent of the of the underlying spaces (although sometimes it does matter).
Your example, by definition, P = Prob(X = 6ft AND Y = raining) = mu{t: X(t) = 6ft and Y(t) = raining}. You have to assume their joint probability space. For example, maybe they are independent, and then it P = Prob(X = 6ft) \*Prob(Y = raining), or maybe it’s Y = if X = 6ft than raining else not raining, and then P = Prob(X = 6ft).
If you don’t have a given joint pobability space, you implicitly construct it (for example, by saying RV are independent, you implicitly construct a product space). Generally, the fact that sometimes you talk about X living on one space (on its own) and other time on the other (joint with some Y) doesn’t really matter, because in most situations, probability theory is specifically about the properties of random variables that are independent of the of the underlying spaces (although sometimes it does matter).
Your example, by definition, P = Prob(X = 6ft AND Y = raining) = mu{t: X(t) = 6ft and Y(t) = raining}. You have to assume their joint probability space. For example, maybe they are independent, and then it P = Prob(X = 6ft) \* Prob(Y = raining), or maybe it’s Y = if X = 6ft than raining else not raining, and then P = Prob(X = 6ft).