Multi variable summation/union

Question

I've seen a process like this in a proof: $$\sum_{k,l}|I_{k,l}|=\sum_k\sum_l|I_{k,l}|$$ and it says we can't do this if the summands are not positive numbers. I thought the left side is an abbreviation of the right side, but something's more in this I guess.

What exactly is $\sum_{k,l}$? and $\cup_{k,l}$? How should I understand these?

Answer 1

My guess is that the observation refers to infinite sums. If $k,l$ are in a finite set of indexes, then $$ \sum_{k,l}a_{i,k}=\sum_k\sum_l a_{k,l}. $$ But this is no longer true if if $k,l$ move in infinite sets of indexes. For instance, $$ \sum_{k,l=0}^\infty a_{i,k}=\sum_{k=0}^\infty\sum_{l=0}^\infty a_{k,l} $$ is guaranteed to hold if one the following conditions holds:

1. $a_{i,j}\ge0$.
2. $\sum_{k,l=0}^\infty |a_{i,k}|<\infty$.