144 1. Measure theory We can express F (b) F (a) as a telescoping series F (b) F (a) = k j=1 F (tj) F (tj−1). To estimate the size of this sum, let us first consider those j for which tj E. Then, by construction, the intervals (tj−1,tj) are disjoint in U. By construction of κ, we thus have j:tj ∗∈E |F (tj) F (tj−1)| ε and thus j:t∗∈E j F (tj) F (tj−1) = O(ε). Next, we consider those j for which tj E. By construction, for those j we have F (tj) F (tj ∗) = (tj tj ∗)F (tj ∗) + O(ε|tj tj ∗|) and F (tj ∗) F (tj−1) = (tj tj−1)F (tj ∗) + O(ε|tj tj−1|) and thus F (tj) F (tj−1) = (tj tj−1)F (tj ∗) + O(ε|tj tj−1|). On the other hand, from construction again we have [tj−1,tj ] F (y) dy = (tj tj−1)F (tj ∗) + O(ε|tj tj−1|) and thus F (tj) F (tj−1) = [tj−1,tj ] F (y) dy + O(ε|tj tj−1|). Summing in j, we conclude 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 the [tj−1,tj] with tj E. By construction, this set is contained in [a, b] and contains [a, b]\U. Since U |F (x)| dx ε, we conclude that S F (y) dy = [a,b] F (y) dy + O(ε). Putting everything together, we conclude that 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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