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Basic Measure Theory Question

  1. May 8, 2012 #1
    1. The problem statement, all variables and given/known data
    My question is would I be allowed to say,
    if lf+-[itex]\phi[/itex]l<ε/(2[itex]\mu[/itex](E)
    then ∫E lf+-[itex]\phi[/itex]l<ε/2

    2. Relevant equations

    E is the set in which we are integrating over.
    [itex]\mu[/itex] is the measure
    [itex]\varphi[/itex] is a simple function
    f+ is the non-negative part of the function f.

    3. The attempt at a solution

    I can't think of any reason that this wouldn't be true but my text is very vague in this chapter and so I am really not sure if this is an ok statement.



    Thank you for your time.
     
  2. jcsd
  3. May 8, 2012 #2

    Fredrik

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    Staff Emeritus
    Science Advisor
    Gold Member

    Yes, your conclusion is correct.

    It's not hard to prove the following:
    ##\newcommand{dmu}{\operatorname{d}\!\mu}##

    (a) If ##f\geq 0## a.e., then ##\int f\dmu \geq 0##.
    (b) If ##f\geq g\geq 0## on E, then ##\int_E f\dmu\geq\int_E g\dmu##.

    Hint: To prove (b), use (a) and the fact that the assumption implies that ##f\chi_E\geq g\chi_E## (everywhere, and therefore a.e.).
     
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