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Math Help - Analysis: Absolute Continuity & Differentiation

  1. #1
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    Analysis: Absolute Continuity & Differentiation

    Let A  \subset [0, 1] be a Borel set such that 0 < m(A  \cap I) < m(I) for all interval I  \subset [0, 1]. Let
    F(x) = m([0, x]  \cap A), where m is lebesgue measure. Show that:
    1. F(x) is absolutely continuous and strictly increasing on [0, 1]
    2. F'(x) = 0 on a set of positive measure.

    I've shown 1, but can't prove 2. In fact, 2 would seem to be false because if F is Abs Cont and F' is zero on a set E, then  F = \int_E F' implies that F is 0 on the set E. But, wouldn't this contradict the fact that F is strictly increasing?
    Last edited by southprkfan1; April 21st 2010 at 08:30 PM.
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  2. #2
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    Quote Originally Posted by southprkfan1 View Post
    Let A  \subset [0, 1] be a Borel set such that 0 < m(A  \cap I) < m(I) for all interval I  \subset [0, 1]. Let
    F(x) = m([0, x]  \cap A), where m is lebesgue measure. Show that:
    1. F(x) is absolutely continuous and strictly increasing on [0, 1]
    2. F'(x) = 0 on a set of positive measure.

    I've shown 1, but can't prove 2. In fact, 2 would seem to be false because if F is Abs Cont and F' is zero on a set E, then  F = \int_E F' implies that F is 0 on the set E. But, wouldn't this contradict the fact that F is strictly increasing?
    Got the answer from another source

     F(x) = \int_{[0,x]} \chi_A(x)dm -->  F'(x) = \chi_A(x)

    So, F'(x) is 0 on [[0,1]\A], which has positive measure.
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