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Assume that $f$ is in $L^p(B,\mu)$, for every $p>1$, where $\mu$ is a measure in a ball $B$ of Euclidean space. Let $I_p=(\int_B (|f|^p)d\mu)^{1/p}$. I would like to find an easy reference of the following formula $$\lim_{p\to \infty} I_p = \|f\|,$$ where $\|f\|=\mathrm{ess\,sup} |f(x)|$.

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migrated from mathoverflow.net Aug 18 '13 at 13:44

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marked as duplicate by Amzoti, Seirios, Sasha, Cameron Buie, Hagen von Eitzen Aug 20 '13 at 6:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
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Dear @user36162: This question will probably receive more attention at math.stackexchange. In general, if your question is about mathematics, but not research level, please consider posting it at math.stackexchange instead. –  Ricardo Andrade Aug 18 '13 at 9:51
    
This was already asked and answered several times on MSE: math.stackexchange.com/q/469212 math.stackexchange.com/q/92147 –  Did Aug 18 '13 at 14:07
    
Besides Wikipedia, a reference can be a real analysis textbook; e.g., G.B. Folland, Real Analysis, 2nd ed., p.187. –  user Aug 20 '13 at 4:25