Suppose I want countable many $X_i$ that are iid to e.g the Gamma distribution $\Gamma(k,\theta)$. How do I actually construct these, given the usual definitions of random variables/measureable functions? Do I just take pairwise disjoint copies of $\mathbb R$ as the domains of the $X_i$ and go from there?
This question came up while I was trying to formalize distributions.