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Possible Duplicate:
Lp space convergence

This is related to a previous question that I posted. Here is the problem:

Suppose that $\{f_n\}$ is a sequence of functions on $[0,1]$ such that $f_n(x) \rightarrow f(x)$ almost everywhere. Suppose also that $\sup{\lVert f_n\rVert_{L^4}} = M < \infty $.

Prove that $\lVert f_n - f\rVert_{L^3} \rightarrow 0$.

I could complete this proof if I knew that $\lVert f_n\rVert_{L^3} \rightarrow \lVert f\rVert_{L^3}$ by a problem that I proved from Royden's book. How can we recover this fact from the problem statement?

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    Remark. It should be noticed that \sup ||f_n||_{L^4}<\infty is important in the proof sketch. Because, it leads to ||f||_4<\infty so that $||f_n-f||_4=\mbox{limit}_{n\rightarrow\infty}||f_n-f||_{L^4([0,1]-N_{\delta_n})}$. That is, it guarantees the uniform continuousness of integration with respect to integral domain. If \sup ||f_n||_{L^4}<\infty fails, then one can easily give a counterexample to show $||f_n-f||_{L^4(N_\delta)}$ can be $\infty$ all the time, which is a lot greater than $||f_n-f||_{L^4([0,1]-N_\delta)}$.2012-08-18

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