Absolute Continuity Problem from Royden

From: mr0x (mr00xx_at_nospam.hotmail.com)
Date: 09/04/04 

Date: Sat, 04 Sep 2004 22:47:58 GMT

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?
Thanks.

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