# Absolute continuity of increasing functions on an interval

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 $$\sum_{k=1}^\infty |f(b_k)-f(a_k)|<\epsilon$$

• The fact that $f$ is increasing means that f maps intervals to intervals – user25959 Nov 24 '18 at 16:36
• What about discontinuities f may have? Since f is increasing it would only have jump discontinuities. Although the endpoints $a_k$ and $b_k$ may be close to one another, how can we ensure that |$f(a_k)-f(b_k)$| can be made small if |$b_k-a_k$| is small enough? I can see how the proof would work in the case f were continuous. – ShaftSinker Nov 24 '18 at 17:36
• Sorry- I had in mind a continuous function. Perhaps it is possible to use the fact that measurable increasing functions have at most countably many discontinuities (jump discontinuities in this case). – user25959 Nov 24 '18 at 17:41
• Can you also show the preceding problem? – Alex Vong Nov 24 '18 at 21:19
• maybe the question should have been, for all measurable $E$, $m^*(E) < \epsilon$ if $m( f^{-1}(E)) < \delta$? (ps Q38 but not Q39 is in the errata www2.math.umd.edu/~pmf/docs/Real%20Analysis.pdf ) – Calvin Khor Nov 25 '18 at 0:55