Fubini’s theorem for sequences says:
If $\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{mn}|<\infty$, then
$\sum_{m=1}^\infty \sum_{n=1}^\infty a_{mn}=\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{mn}|$
My question is then, what is the equivalent of this towards random matrices?
If we have $\sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn}$, where $A_{mn}$ are $d\times d$-random matrices, in which case can we rearrange the infinite sum?
Maybe when: $\sum_{n=1}^\infty \sum_{m=1}^\infty ||A_{mn}||<\infty$ and $||\cdot||$ defines a norm?
Or $\mathbb{E}[\sum_{n=1}^\infty \sum_{m=1}^\infty ||A_{mn}||]<\infty$?