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Let $A, B$ be two sets and take $A \times B$ be the Cartesian product set. I am wondering under what conditions on $A,B$ that the following equality always holds or not:

$$\sum_{(x,y) \in A \times B} f(x,y) = \sum_{x \in A} \sum_{y \in B} f(x,y). $$

I found a link https://proofwiki.org/wiki/Summation_over_Cartesian_Product_as_Double_Summation that closely related to this problem but it requires that $A, B$ are finite sets. Can it be all countable sets? can it be all uncountable sets, say $A=B=\mathbb{R}$ or $A=\mathbb{R}$? I fail to see why not. Any help is appreciated.

Fianra
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For countable sets the question is subtle. The convergence of a countably infinite sum can depend on the order of summation. For positive terms, if either side converges then so will the other, to the same value.

For uncountable sums of positive terms convergence requires that all but countably many terms are $0$.

The sum of an uncountable number of positive numbers

Ethan Bolker
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