=+12.10. Of minor interest is the k-dimensional analogue of (12.4). Let I, be (0, 1] for 1 20 and (1, 0] for 1 ≤ 0, and let A, = / ,, x . .. x/, . Let (x) be +1 or -1 according as the number of i, 1 sis k, for which x; < 0 is even or odd. Show that, if F(x) = (x)u(A,), then (12.12) holds for bounded rectangles A.

Call F degenerate if it is a function of some k - 1 of the coordinates, the requirement in the case k = 1 being that F is constant. Show that A F = 0 for every bounded rectangle if and only if F is a finite sum of degenerate functions; (12.12) determines F to within addition of a function of this sort.