Re: Absolute continuity, another question,

The main issue is how are we to do anything with the integral
\int_a^x g ' (f(t)) f ' (t) dt .

We would want to show this is just the Riemann-Stieltjes integral
\int_a^x g ' (f(x)) d f(x),

which in turn is the Riemann integral

\int_f(a)^f(x) g'(t) dt,

which of course is g(f(x)) - g(f(a)).

Proving these steps is much more elegant in the more general setting of
measure theory. This is mainly because one rarely needs to remember how
to spell Stieltjes.

.