When adding up two random variable with probability space $(\Omega_1, B_1, P_1)$ and $(\Omega_2, B_2, P_2)$, it seems that a new probability space $(\Omega', B', P')$ is implicitly defined by having a surjective function $\pi:\Omega'\mapsto\Omega_1$ which is measurable and $P'(\pi^{-1}(E)) = P_1(E)\;\; \forall E \in B_1$. In this Terrance Tao's note, it shows how one can extends a sample space from 'throwing one dice' to 'throwing two dice' by explicitly defining $\pi$, $\Omega'$ and $B'$.

But in a lot of textbooks, when random variables are added, multiplied or divided together, these are not defined explicitly. A frequently seen example is the Binomial Distribution (from here, but I have seen the same thing in my textbook as well):

Let $X_1$, $X_2$ ..., $X_n$ be identical Bernoulli Distributed random variables with $P(X=1) = p;\; P(X=0) = 1-p$. And their corresponded Binomial random variable is

$X = X_1 + X_2 + ... + X_n $

Here it seems to me that the above equation magically defined $\Omega'$ to be the Cartesian product of the sample spaces of all the $X_i$, and it also defined the probability of each sample points in the new $\Omega'$.

How does the addition formula uniquely determine the new $\Omega'$ and $P'$? In general, what are the rules?

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    $\begingroup$ Remember that, in the end, random variables are just functions, hence one can only add random variables defined on the same probability space and no new implicit probability space is involved. You might want to explain how you envision to add two random variables with respective probability spaces $(\Omega_1, B_1, P_1)$ and $(\Omega_2, B_2, P_2)$. $\endgroup$ – Did Sep 18 '16 at 20:26
  • $\begingroup$ I understand that strictly formally, two random variables with different sample spaces can't be added together. Ideally we should define a good sample space at the very beginning, but in reality people don't. Like, in the example of Binomial, when people firstly introduce a Bernouli variable, they usually just assume a sample space of ${H,T}$, or ${0,1}$. And then when the Binomial is introduced they would suddenly add them together as if they are allowed to do so without any explanation. So my question is how should we interpret when we see this? Is there any rules behind this convention? $\endgroup$ – Haochi Kiang Sep 18 '16 at 20:49
  • $\begingroup$ And also notice that the Bernouli random variables $X_i$'s can have different sample space $\endgroup$ – Haochi Kiang Sep 18 '16 at 20:50
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    $\begingroup$ Introducing ad hoc sample spaces and modifying them each time a new random variable is added to the pot seems to be a popular approach in some curricula. However it is deeply misguided, more or less for the reasons you delineate. A more cogent approach is to know once and for all that sample spaces exist, that are sufficient to manipulate the random variables one is interested in, and to avoid choosing one of them since anyway the choice is irrelevant to every actual computation one can be interested in. $\endgroup$ – Did Sep 18 '16 at 21:01

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