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!

A caracterisation of f=0 by integrals

  1. Apr 7, 2008 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    [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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

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

Have something to add?