Fubini theorem for sequences

Solution 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:

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