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I am reading a lecture note here and it says the following:

If we have a doubly indexed family of nonnegative numbers $a_{ij}, \ i, \ j \in \mathbb{N}$ and if either (i) the numbers are nonnegative, or (ii) the sum $\sum_{(i,j)} a_{ij}$ is absolutely convergent, then

$$ \sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij} = \sum_{j=1}^\infty \sum_{i=1}^\infty a_{ij} = \sum_{(i,j) \in \mathbb{N}^2} a_{ij}$$

More important, we stress that the first equality need not hold in the absence of conditions (i) and (ii).

I am not sure why the last comment holds regarding the first equality where we change the order of the summation. Is there any well known counter-example?

joriki
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melatonin15
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  • I removed the [tag:probability-theory] tag. Please avail yourself of the tag summaries when choosing tags. – joriki Mar 13 '20 at 19:29
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    There seems to be a redundant condition: "a family of nonnegative numbers $a_{ij}$" must always satisfy condition (i). – Michael Mar 13 '20 at 19:48
  • See my example at https://math.stackexchange.com/questions/2006993/conditions-for-series-rearrangement/2007071#2007071 – zhw. Mar 13 '20 at 19:53

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