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Stuck with this problem from Zgymund's book.

Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1. Assume that $\|f_{n}\|_{p} \leq M < \infty$. Prove that:

$\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\dfrac{1}{p} + \dfrac{1}{q} = 1$.

Right, so I estimate the difference of the integrals and using Hölder end up with:

$$\left|\int f_{n} g - \int fg\right| \leq \|g\|_{q} \|f_{n} - f\|_{p}$$

From here I'm stuck because we are not assuming convergence in the seminorm but just pointwise convergence almost everywhere. How to proceed?

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    @Sivaram: but by pointwise convergence almost everywhere don't they mean convergence with respect the Euclidean norm (and not the seminorm?) perhaps that's my mistake. Here $f,f_{n}$ are real-valued so it makes sense to take the Euclidean norm.2010-11-20
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    Take the witch hat (triangle) for $f_n$, $f = 0$. Take $g = 1$ on $[0,1]$ and $0$ otherwise, then $\int f_n g = \int f_n = 1 \not \to \int f = 0$, what am I missing? I think you need to get a dominating function from somewhere.2010-11-20
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    I am sorry. I am wrong. Almost sure convergence doesn't mean convergence in $p^{th}$ mean. Sorry for my wrong comment.2010-11-20
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    Isn't $\|f_n\|_p<\infty$ redundant? It's already given that $f_n\in L^p$. Do you mean $\sum_n \|f_n\|_p<\infty$2010-11-20
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    @Jonas: Don't know what's wrong either, if you want you can check page 144, problem 13.2010-11-20
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    @AgCl: right, just typed it directly from the book but it is redundant, yes.2010-11-20
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    @user10: What book by Zygmund?2010-11-20
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    @user10: I looked it up. It should be $\|f_n\|_p \leq M < \infty$. Could you correct it please?2010-11-20
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    @Jonas: Measure and Integral: An Introduction to Real Analysis.2010-11-20
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    Still, if it is uniformly bounded why does my "counterexample" not work?2010-11-20
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    I am not exactly sure adding $||f_n||_p \leq M$ helps2010-11-20
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    It would be of help if $|f_n| \leq M$. In fact, it would then make the problem a bit trivial.2010-11-20
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    The OP asked 25 questions with all being answered, but accepted none of these answers!2011-12-09

4 Answers 4