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I want to find a counter example

This is the Fubini theorem for sequences:

If $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}|a_{mn}|<\infty,$$

then

$$\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}a_{mn}=\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}a_{mn}.$$

Then, does there exist a sequence $\{a_{mn}\}$ such that the left and the right hand sides of equality are finite but are not equal?

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1 Answer 1

Fubini's theorem (in the special case of sequences) says that if $\sum\sum|a_{mn}|<\infty$ then interchanging the order of summation in $\sum_m \sum_n a_{mn}$ is acceptable. Note the lack of modulus signs here. In contrast, interchanging the order of summation is always acceptable in $\sum\sum |a_{mn}|$, since the summands are nonnegative (this is sometimes called Tonelli's theorem). In this case, equality means that if either side is finite then they both are and they're equal.

If $\sum\sum|a_{mn}|=\infty$, then it may happen that $\sum_m \sum_n a_{mn}$ and $\sum_n \sum_m a_{mn}$ are different, even if they are both finite. Indeed, try summing the rows and columns of the following infinite matrix:

$$\left(\begin{array}{cccccc} 1 & - 1 & 0 & 0 & 0 &\cdots\\ 0 & 1 & -1 & 0 & 0 &\cdots\\ 0 & 0 & 1 & -1 & 0 &\cdots\\ 0 & 0 & 0 & 1 & -1 &\cdots\\ 0 & 0 & 0 & 0 & 1 &\cdots\\ \vdots & \vdots & \vdots &\vdots &\vdots & \ddots\\ \end{array}\right)$$

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Yes, but this doesn't satisfy the condition $\sum^{\infty}_{n=1}\sum^{\infty}_{m=1}|a_{mn}|<\infty$, does it? –  Old John Sep 10 '12 at 14:37
    
@JohnSenior: Such an example is going to be hard to find, as Sean now points out (correctly) in the first paragraph. –  t.b. Sep 10 '12 at 14:49
    
Yes, but I was commenting on Sean's answer before he corrected/explained it :) The initial version seemed to indicate that the example satisfied all the conditions required by the OP. –  Old John Sep 10 '12 at 14:51
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I added the absolute values in the second formula of the question, given that the OP missed it at all (he simply write $\sum_m\sum_n=\sum_n\sum_m$). –  enzotib Sep 10 '12 at 14:55
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@enzotib And you should not have. –  Did Sep 10 '12 at 15:32

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