146 1. Measure theory (ii) If x E, we let δ(x) 0 be a small enough number such that |F (y)−F (x)−(y−x)F (x)| ε|y−x| holds whenever |y−x| δ(x), and such that | 1 |I| I F (y) dy −F (x)| ε whenever I is an interval containing x of length at most δ(x), exactly as in the proof of Theorem 1.6.40. Applying Cousin’s theorem, we can find a partition a = t0 t1 . . . tk = b with k 1, together with real numbers tj [tj−1,tj] for each 1 j k and tj tj−1 δ(tj ∗). As before, we express F (b) F (a) as a telescoping series F (b) F (a) = k j=1 F (tj) F (tj−1). For the contributions of those j with tj E, we argue exactly as in the proof of Theorem 1.6.40 to conclude eventually that j:t∗∈E j F (tj) F (tj−1) = S F (y) dy + O(ε(b a)), where S is the union of all [tj−1,tj] with tj E. Since [a,b]\S |F (x)| dx m=1 Um |F (x)| dx ε, we thus have S F (y) dy = [a,b] F (y) dy + O(ε). Now we turn to those j with tj E. By construction, we have F (tj) F (tj−1) = (tj tj−1)F (tj ∗) + O(ε|tj tj−1|) for these intervals, and so j:tj ∗∈E F (tj) F (tj−1) = ( j:tj ∗∈E (tj tj−1)F (tj ∗)) + O(ε(b a)). Next, for each j we have F (tj ∗) 2m and [tj−1,tj] Um for some natural number m = 1, 2,..., by construction. By countable additivity, we conclude that ( j:t∗∈E j (tj tj−1)F (tj ∗)) m=1 2mm(Um) m=1 2mε/4m = O(ε). Putting all this together, we again have F (b) F (a) = [a,b] F (y) dy + O(ε) + O(ε|b a|). Since ε 0 was arbitrary, the claim follows.
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