Interchanging the Order of Summation for Nets

Question

Suppose we have a net of real numbers $\{x_{(\alpha,\beta)}\}_{I\times J}$ and let's assume $x_{\alpha,\beta}\ge 0$ for every pair of $(\alpha,\beta)\in I\times J$. Does the equalities $$\sum_{(\alpha,\beta)\in I\times J}x_{(\alpha,\beta)} = \sum_{\alpha \in I}\sum_{\beta \in J}x_{(\alpha,\beta)} = \sum_{\beta \in J}\sum_{\alpha \in I}x_{(\alpha,\beta)}$$ always follow?

Answer 1

No. You could have the $2$ pairs $(\alpha, \beta)$ and $( \gamma, \delta)$ ... this does not mean that you have the pair $(\alpha,\delta)$.