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Homework Help: A caracterisation of f=0 by integrals

  1. Apr 7, 2008 #1

    quasar987

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    [SOLVED] A caracterisation of f=0 by integrals

    1. The problem statement, all variables and given/known data
    Does anyone know how to show, or know a book that proves the implication

    [tex]\left(\int_0^1f\varphi = 0 \ \ \forall \varphi \in C_c^1([0,1])\right)\Rightarrow f=0[/tex]

    for f in L²([0,1]) and where [itex]C_c^1([0,1])[/itex] denotes the C^1([0,1]) functions whose support is contained in (0,1).

    Thanks.


    3. The attempt at a solution

    I tried using the density of [itex]C_c^1([0,1])[/itex] in L^1 to obtained a sequence [itex]\varphi_n[/itex] that converges pointwise a.e. to the caracteristic function of [0,1] and then plugging-in the convergence theorems (Fatou, motone and dominated) but I eventually aknowledged that this would not work. Well, at least I got

    [tex]\int_0^1f\leq 0[/tex]

    out of Fatou.
     
  2. jcsd
  3. Apr 7, 2008 #2

    morphism

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    Just to be clear, isn't [itex]C_c^1([0,1])[/itex] the set of compactly supported continuously differentiable funtions on [0,1]?

    Anyway, here are some thoughts. Consider an open subset U of [0,1], whose characteristic function is [itex]\chi[/itex]. Get a sequence [itex]\{\varphi_n\}[/itex] in [itex]C_c^1([0,1])[/itex] that converges a.e. to [itex]\chi[/itex]. We can assume that [itex]M = \sup_n \| \varphi_n \|_\infty < \infty[/itex] (just construct them properly). So [itex]f\varphi_n \to f\chi[/itex] a.e., and [itex]\{f\varphi_n\}[/itex] is majorized by M|f|, which is integrable. Hence, by the dominated convergence theorem, [itex]\int_U f = 0[/itex] for all open subsets U of [0,1]. Now use the regularity of the Lebesgue measure to conclude that f=0 a.e.
     
  4. Apr 7, 2008 #3

    quasar987

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    Dunno if this is standard convention or not but my professor uses [itex]C_c^1([0,1])[/itex] to denote the set of compactly supported continuously differentiable funtions on (0,1). So for instance, the characteristic function of [0,1] is not in [itex]C_c^1([0,1])[/itex] according to my prof.

    Neat argument!

    Say, have you seen my other similar question ?

    https://www.physicsforums.com/showthread.php?t=226834
     
    Last edited: Apr 7, 2008
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