I was wondering whether there is any reference that deals with the distributive law for infinitely many elements, i.e.
$$ \prod_{i\in \mathbb N} \sum_{k\in \mathbb N} \alpha_{i,k} = \sum_{(k_i)_{i\in \mathbb N} \in \mathbb N^\mathbb N} \prod_{i\in \mathbb N} \alpha_{i,k_i}. $$
Unfortunately, I could not find a good reference where this general case is treated and I am pretty sure that there are lots of things that can go wrong here (similar to the rearrangement theorems for series.)
I became interested in this question because I wanted to use
$$ \prod_{i\in \mathbb N} \sum_{k \in \Omega} \alpha_{i,k} = \sum_{(k_i)_{i\in \mathbb N} \in \Omega^\mathbb N} \prod_{i\in \mathbb N} \alpha_{i,k_i}. $$
for $\Omega$ finite.
So if you we can say more about this case, I'd be interested, too.
By the way: Since I am only interested in the case where $\alpha_{i,k}$ are positive numbers, I do not care about the summation order here in my question, which is why I use this sloppy notation where I sum over a set without specifying an order. Please feel free to be more general in an answer.
