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A question about lebesgue integral

  1. Nov 2, 2013 #1
    if lebesgue integral of f^2 over an interval equal 0, must f=0 a.e on that interval?
     
  2. jcsd
  3. Nov 2, 2013 #2
    No. Try to find a counterexample (hint: the integral can be 0 since positive and negative parts cancel out).
     
  4. Nov 2, 2013 #3

    D H

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    What negative parts, R136a1? He's integrating f(x)2 over some interval.


    Is f a function that maps reals to reals, or something else?
     
  5. Nov 2, 2013 #4
    Oh god. Never mind my reply.
     
  6. Nov 2, 2013 #5
    For any measure space [itex](X,\mathcal{S},μ)[/itex], and any measurable function [itex]g:\rightarrow [-∞,∞][/itex], [tex]∫|g|dμ=0\implies g=0 a.e.[/tex]

    Specifically, since [itex]f^2=|f^2|[/itex], this gives [itex]f^2=0[/itex] a.e., and hence [itex]f=0[/itex] a.e.
     
  7. Nov 2, 2013 #6

    D H

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    You are assuming f is a real function, Axiomer. If it's a complex function, then f2 is not the same as |f2|.
     
  8. Nov 2, 2013 #7
    That's a good point. Since the op didn't specify otherwise, I assumed we were talking about functions to the extended real line.
     
  9. Nov 2, 2013 #8
    For any measure space [itex](X,\mathcal{S},μ)[/itex], and any measurable function [itex]g:\rightarrow [-∞,∞][/itex], [tex]∫|g|dμ=0\implies g=0 a.e.[/tex]

    proof:
    Define [itex]A=\{x\in X: g(x)≠0\}[/itex]. For all naturals n, define [itex]A_n=\{x\in X: |g(x)|>\frac{1}{n}\}[/itex].

    [itex]\frac{1}{n}μ(A_n)=∫\frac{1}{n}x_{A_n}dμ≤∫|g|dμ=0[/itex], so [itex]μ(A_n)=0[/itex] for all n.

    Then [itex]μ(A)=μ(\bigcup _{n=1}^∞A_n)≤\sum _{n=1}^∞μ(A_n)=0\implies μ(A)=0[/itex], as desired.
     
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