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Finite Measure Space Problem

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    I have a sequence of functions converging pointwise a.e. on a finite measure space, [tex] \int_X |f_n|^p \leq M (1 < p \leq \infty [/tex] for all n. I need to conclude that [tex] f \in L^p [/tex] and [tex] f_n \rightarrow f [/tex] in [tex] L^t [/tex] for all [tex] 1 \leq t < p. [/tex]


    2. Relevant equations



    3. The attempt at a solution
    By Fatous I can show [tex] f \in L^p [/tex] and since [tex] L^t \subseteq L^p [/tex] for finite measure spaces, I have everything in L^t as well. I can apply Egoroffs to get [tex] \int_E |f_n-f|^t < \epsilon [/tex] with [tex] \mu(X-E) < \delta [/tex]. Any ideas on how to proceed? And thanks for your time!
     
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  3. Nov 12, 2011 #2

    micromass

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    Maybe try something like

    [tex]\int_X{|f_n-f|^t}=\int_{X\setminus E}{|f_n-f|^t}+\int_{E}{|f_n-f|^t}[/tex]

    Try to find an upper bound K of [itex]|f_n-f|^t[/itex]. Then the first integral becomes

    [tex]\int_{X\setminus E}{|f_n-f|^t}\leq \mu(X\setminus E) K\leq \varepsilon K.[/tex]
     
  4. Nov 12, 2011 #3
    Thanks for the reply!
    This is what I've been trying, but I cannot see a reason why |f_n-f|^t should be bounded on this set.
     
  5. Nov 12, 2011 #4

    micromass

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    OK, here's how to proceed:

    First, prove that

    [tex]\lim_{\lambda \rightarrow 0}{\ \ \sup_n{\int_{\{|f_n|^t\geq \lambda\}}{|f_n|^t}}}=0[/tex]

    Hint: if [itex]0<a<b[/itex], then [itex]b^t=b^{t-p}b^p\leq a^{t-p}b^p[/itex]

    Second, prove that for each [itex]\varepsilon >0[/itex], there exists a [itex]\delta>0[/itex] such that for each E with [itex]\mu(E)<\delta[/itex], we have

    [itex]\int_E{|f_n|^t}<\varepsilon[/itex]

    for each n.
     
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