Can we define sum over an uncountable set?

Question

Is it possible by some means to define a notion of sum over the elements of an uncountable set $S$ of real numbers.I thought of something like $\sum_{\alpha \in \tau } a_\alpha:=\sup_{T\subset S ,|T|=\aleph_0}\sum_{a_{\alpha}\in T}a_\alpha$ where $\tau$ is the index set.Does this makes much sense?