the problem is, A and B are not necessarily lebesque measurable, which, if they were, the disjointness would give the desired result. but the outer measure is defined on the entire power set of R, that is, on sets that are not necessarily lebesque measurable also. the proof will have something to do with infs and covering sets, but i just cant seem to work it out yet.
One inequality (
) is trivial. To get the other inequality, since A,B are separated, there are disjoint opens U and V that contain A and B respectively. Take any covering
(by (disjoint) Borel subsets or by (disjoint) intervals, depending on how you define the Lebesgue outer measure), and you have covering
of A and
of B, giving