The final inequality looks incorrect to me. First, there is no reason for the inclusion of boundary points from the rectangles S to increase the integral (consider a negative f). Second, it seems to me like he wants to say that the final two expressions are equal, but that sort of lemma has not yet been proven, so we don't know that the image of the boundary of the rectangle S under the map g is measure zero. Am I totally missing something? Is this special case of the usual lemma (g a c-1 function and S a rectangle in R^n) supposed to be obvious?
