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This was on an old qualifying exam. I'm not sure really where to start. I know the proof and statement of Vitali's convergence theorem, and I thought it might work here in some fashion?

Let $(X,M,μ)$ be a measure space, with $μ(X)<∞$ . Let $f_n$ be a sequence of essentially bounded functions. Suppose that $\sup{||f_n||}_∞$ is finite. Also, suppose that $f_n→f$ almost everywhere. For which p is the following statement true : "If f is in $L_p(μ)$, then $||f_n−f||_{L_p}→0$"

I also know the following is true, proving it as a problem in Royden: If $f_n \in L_p$ and $f_n \rightarrow f $ almost everywhere, then $||f_n||_{L_p} \rightarrow ||f||_{L_p}$ if and only if $||f_n -f||_{L_p} \rightarrow 0 $.

Any help you can give is great. Thanks in advance.

James
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1 Answers1

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You don't need much machinery for this problem.

  1. Show that $f$ is essentially bounded.

  2. For $p < \infty$, you need to show $\int_X |f_n - f|^p\,d\mu \to 0$; use dominated convergence.

  3. For $p=\infty$, look for a counterexample.

Nate Eldredge
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  • I'm sorry, I must be thinking of this incorrectly. For 1 : Each $f_n$ is in $L_p$ for every $p$. Thus, we can prove that also $f$ is $L_p$ for every $p$ by Fatou: $\int{|f|^p} = \int{\liminf{|f_n|^p} \leq \liminf{\int{|f_n|^p}} \leq M} $ for some $M$. But I don't get how we can show essentially bounded. – James Aug 18 '12 at 02:52