136 1. Measure theory Given any interval [a, b], we define the total variation F TV ([a,b]) of F on [a, b] as F TV ([a,b]) := sup a≤x0...xn≤b n i=1 |F (xi) F (xi+1)| thus the definition is the same, but the points x0,...,xn are restricted to lie in [a, b]. Thus, for instance, F TV (R) = supN→∞ F TV ([−N,N]). We say that a function F has bounded variation on [a, b] if F BV ([a,b]) is finite. Exercise 1.6.35. If F : R R is a monotone function, show that F TV ([a,b]) = |F (b) F (a)| for any interval [a, b], and that F has bounded variation on R if and only if it is bounded. Exercise 1.6.36. For any functions F, G: R R, establish the triangle property F +G TV (R) F TV (R) + G TV (R) and the homogeneity prop- erty cF TV (R) = |c|F TV (R) for any c R. Also, show that F TV = 0 if and only if F is constant. Exercise 1.6.37. If F : R R is a function, show that F TV ([a,b]) + F TV ([b,c]) = F TV ([a,c]) whenever a b c. Exercise 1.6.38. (i) Show that every function f : R→R of bounded variation is bounded, and that the limits limx→+∞ f(x) and limx→−∞ f(x), are well de- fined. (ii) Give a counterexample of a bounded, continuous, compactly sup- ported function f that is not of bounded variation. Exercise 1.6.39. Let f : R R be an absolutely integrable function, and let F : R R be the indefinite integral F (x) := [−∞,x] f(x). Show that F is of bounded variation, and that F TV (R) = f L1(R). (Hint: The upper bound F TV (R) f L1(R) is relatively easy to establish. To obtain the lower bound, use the density argument.) Much as an absolutely integrable function can be expressed as the dif- ference of its positive and negative parts, a bounded variation function can be expressed as the difference of two bounded monotone functions: Proposition 1.6.34. A function F : R R is of bounded variation if and only if it is the difference of two bounded monotone functions. Proof. It is clear from Exercises 1.6.35, 1.6.36 that the difference of two bounded monotone functions is bounded. Now define the positive variation
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