Measurable set

Given a set and an associated sigma-algebra, (S, \Sigma), a set A \in S is measurable if A \in \Sigma. 
This should be intuitive because a measure tends to be defined on a sigma-algebra?? 
* I think because of the sigma-additivity of a measure, it is necessary to have the countable unions/intersections also be in the set of subsets (like for a sigma-algebra). 
* Do we need the closed under complementation for a measure???