Let $(X, \mu)$ be a measure space and let $f$ be a complex measurable function on $X$. Set $\phi(p)=\int_{X}|f|^pd\mu=||f||_p^p$ for $p\in(0, +\infty)$ and let $E=\{p|\phi(p)<+\infty\}$. I can prove that $E$ is connected. Is $E$ necessarily open? Closed? Can $E$ consist of one point? Can $E$ be any connected subset of $(0,+\infty)$? Thanks a lot.
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https://math.stackexchange.com/questions/264501/can-e-be-a-point?rq=1 – somitra Jun 19 '19 at 06:26
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The answer is yes to all the questions – somitra Jun 19 '19 at 06:27