Question

Show that if there is two sets S1 and S2, S1 and S2 are Jordan regions so is S1 \ (S1 ∩ S2).

Answer 1

Solution. Let C be a Jordan measurable set in a closed rectangle A and fix e > 0. Then there exists a collection of open sets U1, U2, ... which cover aC and such that vol(U:) <e. For every r e ac let U, be a rectangle containing a lying in one of the U;. Since OC is closed and bounded, and hence compact, it is covered by finitely many of these rectangles, Unp,...,U... Hence consider a partition of A\U?=U2, and let Si be those rectangles in the interior of C with S2 = SU {U2,...U.,}. Conversely, given a partition of A and collections S, and S, which satisfy Σvol(S) - Σvol(S) <ε, SES, SES those rectangles lying in Si S2 cover aC and have total volume less that e, hence C is Jordan measurable.