Re: Absolute continuity, another question,

Need to know:
1. Definition of absolute-continuity of (signed) measures. \mu <<
\lambda

2. Definition of the Borel-Stieltjes measure associated with any
measurable continuous on the right function of bounded variation.

3. The Radon-Nikodym derivative.


I. A measurable, continuous on the right function of bounded variation
is absolutely continuous if and only if its associated Borel-Stieltjes
measure is absolutely continuous with respect to Lebesgue measure.

II.The Radon-Nikodym chain rule:
If a << b << c The Radon-Nikodym derivatives satisfy da/dc = da/db *
db/dc almost everywhere.

III. The following change-of-variables formula: (i think this is
right...)

Let x be Lebesgue measure. Let g be integrable, let f be monotone
increasing and absolutely continuous. Let F be the Borel-Stieltjes
measure associated with f.
\int_[a,b] g o f dF= \int_[a,b] g o f (dF/dx) dx = \int_f([a, b]) g
dx = \int_[f(a), f(b)] g dx

This can be proven by letting g = g+ - g-, and producing monotone
sequences of simple measurable functions approaching g+ and g-. Then
apply the monotone convergence theorem.

.