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**1. Homework Statement**

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!