Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Absolute Continuity

  1. Nov 10, 2005 #1
    Let f in AC[0,1] monotonic,Prove that if m(E)=0 then m(f(E))=0
     
  2. jcsd
  3. Nov 10, 2005 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    What are AC, E, and m? Why wouldn't you bother to define these?
     
  4. Nov 10, 2005 #3
    Definitions

    Let f in AC[0,1] monotonic,Prove that if m(E)=0 then m(f(E))=0

    ie, f is absolutely continuous in [0,1], m denotes the Lebesque measure and E is a subset of [0,1] with meausre 0.
     
  5. Nov 11, 2005 #4

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Have you tried anything? For every [itex]\epsilon > 0[/itex], there exists a countable collection of pairwise disjoint open intervals [itex]\mathcal{C}[/itex] such that

    [tex]E \subseteq \bigcup _{U \in \mathcal{C}} U[/tex]

    and

    [tex]\sum _{U \in \mathcal{C}} \mbox{vol}(U) < \epsilon[/tex]

    Absolute Continuity
     
    Last edited: Nov 11, 2005
  6. Nov 11, 2005 #5

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    first try an easier case: let f be lipschitz continuous, i.e. assume there is a constant K such that |f(x)-f(y)| < K|x-y| for all x,y, in domain f.

    then prove f preserves measure zero.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook