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Integrate and measure problem.

Assume $\mu(X)=1$, $f \in L^{p} (X,M,\mu)$ for some $0

I want to prove that: $$\lim_{p\to 0}||f||_p = e^{\int_X \log|f|d \mu}$$

I'm going to prove $\ge$ part using Jensen inequality, but I cannot go opposite side. How can I make it?

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    The discrete (finite) version of this statement can be shown directly by continuity of exponentiation and l'Hospital rule, but I'm not sure how to generalize it to arbitrary spaces.2012-06-14
  • 0
    [The same question here](http://math.stackexchange.com/q/156878/8271)2012-06-14

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