ABSTRACT

P roof. Since \dg\ is integrable over K Theorem 1 (§2.5) yields a function w on K such that dw = \dg\. Define u =

\d{w + g) — \{dw + dg) = \{\dg\ + dg) = (dg)+ . Similarly dv = l(\dg\ - dg) = {dg)~. □

Note that w = u + v and dw > 0, du> 0, dv > 0. So u, v, w are monotone. Hence, a function g of bounded variation on K has finite unilateral limits g(t-) at t in (a, b] and g(t+) at t in [a, b). Also g has at most countably many points of discontinuity.