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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.