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.
2026-05-15 10:41:34.1778841694
Lebesgue integrable nonexample
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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