When a net of positive real numbers converges?

Question

Let $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ be a two sets of non-negative real numbers. If $a_i\leq b_i$ for every $i$, and $\sum_{i\in I}b_i=1$, then when $\sum_{i\in I}a_i$ converges? If $I$ is countable, then it converges. But I want to know that is there a more general result?

Answer 1

An uncountable series converges only if a countable subset of its elements are non-zero. Therefore if $\sum_{i \in I} b_i$ convergences then there exists a countable subset $J$ s.t. if $i \notin J \implies b_i = 0$ and since $a_i \leq b_i$ this implies that $a_i = 0 \: \forall i \notin J$. Therefore the uncountable series $\sum_{i \in I} a_i$ reduces to a countable one and you can apply comparison test of countable series to conclude the series converges.