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I understood the proof but why does this imply that $\{p \in [1, \infty]: f \in L^p\}$ is an interval?

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The inequality implies that if $f\in L^p$ and $f\in L^q$ for $p<q$, then $f\in L^r$ for all $p<r<q$. This implies that the set is an interval.

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