134 1. Measure theory Note that cx is the measure of the interval (F−(x),F+(x)). By mono- tonicity, these intervals are disjoint by the boundedness of F , their union is bounded. By countable additivity, we thus have x∈A cx ∞, and so if we let Jx be the basic jump function with point of discontinuity x and fraction θx, then the function Fpp := x∈A cxJx is a jump function. As discussed previously, G is discontinuous only at A, and for each x A one easily checks that (Fpp)+(x) = (Fpp)−(x) + cx and Fpp(x) = (Fpp)−(x) + θxcx where (Fpp)−(x) := limy→x− Fpp(y), and (Fpp)+(x) := limy→x+ Fpp(y). We thus see that the difference Fc := F −Fpp is continuous. The only remaining task is to verify that Fc is monotone non-decreasing, thus we need Fpp(b) Fpp(a) F (b) F (a) for all a b. But the left-hand side can be rewritten as x∈A∩[a,b] cx. As each cx is the measure of the interval (F−(x),F+(x)), and these intervals for x A [a, b] are disjoint and lie in (F (a),F (b)), the claim follows from countable additivity. Exercise 1.6.33. Show that the decomposition of a bounded monotone non-decreasing function F into continuous Fc and jump components Fpp given by the above lemma is unique. Exercise 1.6.34. Find a suitable generalisation of the notion of a jump function that allows one to extend the above decomposition to unbounded monotone functions, and then prove this extension. (Hint: The notion to shoot for here is that of a “local jump function”.) Now we can finish the proof of Theorem 1.6.25. As noted previously, it suffices to prove the claim for monotone non-decreasing functions. As differ- entiability is a local condition, we can easily reduce to the case of bounded monotone non-decreasing functions, since to test differentiability of a mono- tone non-decreasing function F in any compact interval [a, b] we may replace F by the bounded monotone non-decreasing function max(min(F, F (b)), F (a)) with no change in the differentiability in [a, b] (except perhaps at the endpoints a, b, but these form a set of measure zero). As we have al- ready proven the claim for continuous functions, it suffices by Lemma 1.6.31 (and linearity of the derivative) to verify the claim for jump functions. Now, finally, we are able to use the density argument, using the piecewise constant jump functions as the dense subclass, and using the second part of Lemma 1.6.26 for the quantitative estimate fortunately for us, the density
Previous Page Next Page