1. The problem statement, all variables and given/known data Assume S contained in R2 is bounded. Prove that if S is Riemann measurable, then so are its interior and closure 2. The attempt at a solution Proof: If S is Riemann measurable, its boundary is a zero set. Since the boundary of each open U in the int(S) is part of the boundary of S, this means that the boundary of the int(S) is also a zero set. Since S is bounded, so is its interior. Thus int(S) is Riemann measurable. Since the boundary of S is the closure minus the interior and the boundary of S is a zero set, the closure must also be a zero set. So it is also Riemann measurable. QED. At first I thought this was fine, but then I ran into trouble when I try to prove that if the interior and the closure is Riemann integrable then so is S Any help is appreciated!