142 1. Measure theory (ii) A function F : R R is uniformly continuous if, for every ε 0, there exists a δ 0 such that |F (b) F (a)| ε whenever (a, b) is an interval of length at most δ. Definition 1.6.39. A function F : R R is said to be absolutely continu- ous if, for every ε 0, there exists a δ 0 such that n j=1 |F (bj)−F (aj)| ε whenever (a1,b1),..., (an,bn) is a finite collection of disjoint intervals of to- tal length n j=1 bj aj at most δ. We define absolute continuity for a function F : [a, b] R defined on an interval [a, b] similarly, with the only difference being that the intervals [aj,bj] are of course now required to lie in the domain [a, b] of F . The following exercise places absolute continuity in relation to other regularity properties: Exercise 1.6.49. (i) Show that every absolutely continuous function is uniformly con- tinuous and therefore continuous. (ii) Show that every absolutely continuous function is of bounded vari- ation on every compact interval [a, b]. (Hint: First show this is true for any sufficiently small interval.) In particular (by Exercise 1.6.41), absolutely continuous functions are differentiable almost everywhere. (iii) Show that every Lipschitz continuous function is absolutely contin- uous. (iv) Show that the function x x is absolutely continuous, but not Lipschitz continuous, on the interval [0, 1]. (v) Show that the Cantor function from Exercise 1.6.48 is continuous, monotone, and uniformly continuous, but not absolutely continu- ous, on [0, 1]. (vi) If f : R R is absolutely integrable, show that the indefinite integral F (x) := [−∞,x] f(y) dy is absolutely continuous, and that F is differentiable almost everywhere with F (x) = f(x) for almost every x. (vii) Show that the sum or product of two absolutely continuous func- tions on an interval [a, b] remains absolutely continuous. What happens if we work on R instead of on [a, b]? Exercise 1.6.50. (i) Show that absolutely continuous functions map null sets to null sets, i.e., if F : R R is absolutely continuous and E is a null set, then F (E) := {F (x): x E} is also a null set.
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