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If $\phi:[0,1] \rightarrow (0,\infty)$ is a simple function, please give me a hint for how to calculate this limit: $$\lim_{p\to 0^+}\|\phi\|_p$$

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Hint: start with the simplest case. What happens if $f$ is constant? Try to generalize from there. – mrf Feb 22 at 22:08
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After the constant, then do the case where $\phi$ has two values. Perhaps it will be easier to generalize from that than from the constant case. – GEdgar Feb 22 at 22:14
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I proved a more general case here. – Ayman Hourieh Feb 22 at 22:18
@Ayman Hourieh:Thanks. therefore we have $\lim_{p\to 0^+}||\phi||_p=e^{\int log\phi} $ – rese Feb 22 at 22:45
Yes. Try to prove it where $\phi$ has two values as GEdgar suggested. You'll arrive at this result. If you get stuck, edit your question and we will help. – Ayman Hourieh Feb 22 at 23:13

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