I'm trying to prove, per ex. 5 of section 2.2 of S. Berberian's Fundamentals of Real Analysis, that where [itex]\lambda^*[/itex] is the Lebesgue outer measure, and An is any sequence of (not necessarily measurable) sets of reals increasing to A, then [itex]\lambda^*(A_n)[/itex] increases to [itex]\lambda^*(A)[/itex]. As a hint, it mentions that every set is contained in a measurable set which differs from it by a null set. I considered the closure. I know that the boundary is not necessarily null (for instance, the rationals) but perhaps this cannot be the case for a nonmeasurable set. In either case, assuming the hint, the proof is pretty trivial. I just don't really know where to start on proving the hint.