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Absolutely continuous functions and sets of measure 0.

  1. Oct 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero.

    2. Relevant equations

    A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')} of nonoverlapping intervals in [a,b] with ∑|xj'-xj| < δ, ∑|f(xj')-f(xj)| < ε .



    3. The attempt at a solution

    I think that there is an alternative definition of absolute continuity using countable intervals instead of finite intervals, and if I knew that the set E was countable I think I could go from there...but I don't know that E is countable; I only know that it has measure zero. So I'm not really sure where to start.

    Any help would be much appreciated : )
     
  2. jcsd
  3. Oct 28, 2011 #2
    Well, if you can show that the measure of f(E) is less than any epsilon, you'll be done right? Fix an [itex] \epsilon > 0 [/itex]. Furthermore, you know that E has zero measure, so can you cover it with intervals in a way that allows you to exploit absolute continuity?
     
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