Results 1 to 3 of 3

Math Help - F not absolutely continuous

  1. #1
    Senior Member Dinkydoe's Avatar
    Joined
    Dec 2009
    Posts
    411

    F not absolutely continuous

    I have the following function F:[0,1]\to[0,1] (it is a distr. function. Not so important)

    It satisfies

    F(x) = 1/2 for x\in [1/4,3/4]
    F(1-x) = 1-F(x) for all x
     F(x)= 2F(x/4) for x\in [0,1/4]

    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 \epsilon exists \delta s.t. whenever a disjoint (finite) sequence [x_1,y_1],\cdots [x_n,y_n] of sets with \sum_k|x_k-y_k|<\delta we have \sum_{k}|F(y_k)-F(x_k)| < \epsilon.

    I can't think of any reason why this is...
    Last edited by Dinkydoe; October 16th 2012 at 04:31 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Dinkydoe's Avatar
    Joined
    Dec 2009
    Posts
    411

    Re: F not absolutely continuous

    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2010
    Posts
    980
    Thanks
    236

    Re: F not absolutely continuous

    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 \sum_k|x_k-y_k|<\delta for any given \delta while \sum_{k}|F(y_k)-F(x_k)| is some constant.

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Series Absolutely Convergent, xn/(1+xn^2) absolutely convergent
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 6th 2011, 12:36 PM
  2. Weak convergence of absolutely continuous probability measures
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 18th 2011, 07:36 PM
  3. Uniformly Continuous but not absolutely continuous example?
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: March 9th 2010, 01:23 PM
  4. Uniformly Continuous but not absolutely continuous example?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 9th 2010, 10:43 AM
  5. absolutely continuous functions and measurable sets
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: December 9th 2008, 12:07 PM

Search Tags


/mathhelpforum @mathhelpforum