2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ The fact that $f$ is increasing means that f maps intervals to intervals $\endgroup$ – user25959 Nov 24 '18 at 16:36
  • $\begingroup$ 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. $\endgroup$ – ShaftSinker Nov 24 '18 at 17:36
  • $\begingroup$ 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). $\endgroup$ – user25959 Nov 24 '18 at 17:41
  • 1
    $\begingroup$ Can you also show the preceding problem? $\endgroup$ – Alex Vong Nov 24 '18 at 21:19
  • 1
    $\begingroup$ 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 ) $\endgroup$ – Calvin Khor Nov 25 '18 at 0:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.