You have to define a sigma-algebra (of the measurable sets), to which you apply the Lebesgue measure !
In particular, if the sigma-algebra is the Borel sigma-algebra over [a,b], it's generated by the open subsets of [a,b] (with the usual topology)
And since a closed set is the complement of an open set, it's still in the Borel sigma-algebra, and hence is measurable.
As a side note : if one talks about open sets, there must be a definition of the topology you're using. But we'll just assume it's the usual topology...