Re: Absolute Continuity Problem from Royden

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 09/05/04 

Date: Sun, 05 Sep 2004 06:28:27 -0500

On Sat, 04 Sep 2004 22:47:58 GMT, "mr0x" <mr00xx@nospam.hotmail.com>
wrote:

>Hi, I'm having a lot of problem solving problem 17b from Chapter 5 from
>Royden.
>
>It says
>b).Let E = {x : g'(x)=0}. Then, m(g[E])=0.
>
>g is defined on part (a) of the question.
>
>The (a) of the question is
>a). Let F be absolutely continuous on [c,d] and g be absolutely continuous
>with c <= g <= d on [a,b]. Then, Fog is absolutely continuous on [a,b].
>
>
>I was thinking of a proof like this ->
>Let E' be a subset of E where for each interval in E, we take only 1 point
>from it. Then, m(E')=0 (since it's countable) and so as g[E] = g[E'] and
>thus, m(g[E])=0.
>
>But, on problem 19a), it says ->
>19a). Construct an absolutely continuous strictly monotone function g on
>[0,1] such that g'=0 on a set of positive measure.
>
>Thus, in the proof I can't say m(E')=0 from this.
>
>
>Any hints on what to go about it?

start by noting that m(E) and m(g{E)) are not the
same thing - that should clear up the contradictory
parts...

>Thanks.
>

************************

David C. Ullrich

sorry about the inelegant formatting - typing
one-handed for a few weeks...

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