Ok sorry, it might be important that F is monotone increasing, and F(0) =0, F(1) = 1. Like i said, it's a distribution function...It looks a lot like this: Cantor Function -- from Wolfram MathWorld
I have the following function (it is a distr. function. Not so important)
It satisfies
for
for all
for
I want to show F is not absolutely continuous w.r.t. Lebesgue measure. Should be straightforward but i dont understand...
It is continuous, but not absolutely continuous.
Apparantly, according to the definition it means the following does not hold: For all exists s.t. whenever a disjoint (finite) sequence of sets with we have .
I can't think of any reason why this is...
Ok sorry, it might be important that F is monotone increasing, and F(0) =0, F(1) = 1. Like i said, it's a distribution function...It looks a lot like this: Cantor Function -- from Wolfram MathWorld
I'm not sure I understand how F is constructed. Is it the limit of the process you describe?
Also, I think the definition of absolute continuity allows the sequence of intervals to be countably infinite. Given that, you can probably construct a sequence of intervals with for any given while is some constant.
- Hollywood