I have been using Real Analysis by Royden and Fitzpatrick. I am currently stuck on problem 39 of chapter 6. The problem is stated as follows:
"Use the preceding problem to show that if f is increasing on [a, b], then f is absolutely continuous on [a, b] if and only if for each $\epsilon$ , there is a $\delta$ > 0 such that for a measurable subset E of [a, b], m*(f(E))< $\epsilon$ if m(E) <$\delta$"
(m* denotes outer measure and m denotes lebesgue measure)
I can see why absolute continuity implies the other property. However, I fail to see how absolute continuity follows from the fact that f is increasing and that for each $\epsilon$ , there is a $\delta$ > 0 such that for a measurable subset E of [a, b], m*(f(E))< $\epsilon$ if m(E) <$\delta$"
My approach is to take $\delta$ responding to $\epsilon$. Then for any finite collection of disjoint open interval ($a_k$,$b_k$) such that $\sum_{k=1}^n (b_k-a_k) < \delta$ we have that m*$(f(\cup_{k=1}^n(a_k, b_k)))<\epsilon$. I cannot see how this implies $\sum_{k=1}^n |f(b_k)-f(a_k)|<\epsilon$ as required for absolute continuity.
Is this a good approach to solve this problem? Any advice would be appreciated!
EDIT:
The previous problem (Q38):
Show that f is absolutely continuous on [a, b] if and only if for each $\epsilon$ > 0, there is a $\delta$ > 0 such that for every countable disjoint collection ${(a_k, b_k)}$ of open intervals in (a, b), if $\sum_{k=1}^\infty |b_k-a_k|<\delta$ then $\sum_{k=1}^\infty |f(b_k)-f(a_k)|<\epsilon$