1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    2015 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?