Does $f\in \mathrm L^p$ imply that $f$ is Lebesgue integrable? If not can anyone give me an example?
I don't have any conditions on my domain or the function.
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$1/x\in L^2(1,\infty)$ but not in $L^1$. – Yanko Jun 16 '17 at 14:20
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@yanko But it is in $L^1$ – shvd1991 Jun 16 '17 at 14:23
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No it is not because the sum $\sum \frac{1}{n}$ converge to infinity – Yanko Jun 16 '17 at 14:24
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What is the domain of the functions? If it is some compact interval, then yes, $f\in L^p$ implies $f$ is Lebesgue integrable. – edm Jun 16 '17 at 15:22
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On a domain $\Omega$ with finite measure $L^q(\Omega)$ is contained in $L^p(\Omega)$ for $1\leq p\leq q \leq \infty$. But on unbounded domains there need not be a strict relation like this as shown by the comment by yanko.
You can find some details in here
praveen kr
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So the sufficient condition is that the domain should be bounded. thanks a lot – shvd1991 Jun 19 '17 at 06:40