132 1. Measure theory G(−y) whenever −x −y −a and −x (−b, −a) lies outside of all of the −In. Observe that if x (a, b), and G(−x) G(−y) for all −x −y −a, then D−F (x) r. Thus we see that Er,R is contained inside the union of the intervals In = (an,bn). On the other hand, from the first part of Lemma 1.6.26 we have m(Er,R (an,bn)) F (bn) F (an) R . But we can rearrange the inequality G(−an) G(−bn) as F (bn) F (an) r(bn an). From countable additivity, one thus has m(Er,R) r R n bn an. But the (an,bn) are disjoint inside (a, b), so from countable additivity again, we have n bn an b a, and the claim follows. Remark 1.6.29. Note that if F was not assumed to be continuous, then one would lose a factor of C here from the second part of Lemma 1.6.26, and one would then be unable to prevent D+F from being up to C times as large as D−F . So sometimes, even when all one is seeking is a qualitative result such as differentiability, it is still important to keep track of constants. (But this is the exception rather than the rule: for a large portion of arguments in analysis, the constants are not terribly important.) This concludes the proof of Theorem 1.6.25 in the continuous monotone non-decreasing case. Now we work on removing the continuity hypothesis (which was needed in order to make the rising sun lemma work properly). If we naively try to run the density argument as we did in previous sec- tions, then (for once) the argument does not work very well, as the space of continuous monotone functions are not sufficiently dense in the space of all monotone functions in the relevant sense (which, in this case, is in the total variation sense, which is what is needed to invoke such tools as Lemma 1.6.26.). To bridge this gap, we have to supplement the continuous mono- tone functions with another class of monotone functions, known as the jump functions. Definition 1.6.30 (Jump function). A basic jump function J is a function of the form J(x) := 0 when x x0, θ when x = x0, 1 when x x0, for some real numbers x0 R and 0 θ 1 we call x0 the point of discontinuity for J and θ the fraction. Observe that such functions are monotone non-decreasing, but have a discontinuity at one point. A jump function is any absolutely convergent combination of basic jump functions,
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