From: "David C. Ullrich" <ullrich@math.okstate.edu>
Subject: Re: Absolute Continuity question
Date: Sat, 18 Mar 2000 13:40:00 -0600
Newsgroups: sci.math,sci.math.research
Summary: Inverses of smooth homeomorphisms also absolutely continuous? (No)

Jeff Rubin wrote:

> Suppose f:[a, b]->[c,d] is continuously differentiable, one-to-one,
> and that f(a)=c and f(b)=d. Then f is a homeomorphism and is absolutely
> continuous. Let g:[c, d]->[a,b] be the inverse function of f. Now g
> need not be differentiable. What about absolute continuity? Does g
> have to be absolutely continuous? Why?
> Certainly if f' is never zero, then g is continuously differentiable
> and hence absolutely continuous. But what if f' does have zeroes?

    Pretty sure the answer is no. You can have f strictly increasing
and still have f' = 0 on a set of positive measure - seems likely
that f(that set) will have measure zero, hence g takes a set
of measure zero to a set of positive measure, so g is not
absolutely continuous.

    If this is homework you should stop reading, I've said
too much already. If not:

    Take a=b=c=d=0, I = [0,1]. Let F be a "fat Cantor set":
a subset of I homeomorphic to the traditional middle-thirds 
Cantor set, but of positive measure. Choose a continuous
function phi with phi = 0 on F and phi > 0 on I \ F.
Normalize phi so it has integral 1 and let f be the indefinite
integral of phi.
    Then f is continuously differentiable and the fact that
the complement of F is dense shows that f is strictly
increasing. We need only show that f(F) has measure
zero. So we need only show f(F) has measure less than
epsilon, for any epsilon > 0.
    But if epsilon > 0 we can find an open set V with F
contained in V and such that f' < epsilon on V. (Like
V = the set where f' < epsilon might work.) The point
to switching from F to V here is that V has a very simple
structure: V is just a countable union of disjoint intervals
(could take a finite union of intervals for that matter)
and f' < epsilon on V, hence f(V) is contained in a
union of intervals of total length less than epsilon
times the measure of V, hence less than epsilon.

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