Is double summation distributive?

Question

Is double summation distributive like single summation? For example in the proof for linearity of expectations, $E(ax+by)=aE(x)+bE(y)$, it is shown using double summation and they seemed to break it down using distributive properties. Can someone please confirm?

Answer 1

Yes, as long as each sum is convergent on its own, for example: $$\sum_i\sum_j(a_{ij}+b_{ij})=\underbrace{\sum_i\sum_ja_{ij}}_{S_1}+\underbrace{\sum_i\sum_jb_{ij}}_{S_2}$$ as long as both $S_1,S_2$ converge independently