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This results holds by definition when $p = 1,\infty$. It seems we can use Holder inequality, but I am not figuring out the exact proof.

Thank you.

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Main idea:

For simplicity, let the space be the reals. Let $f\in L^p(\mathbb{R})$. Let $K$ be an arbitrary compact subset. Then $m(K)<\infty$. Hence, $\int_K |f|\leq \|f\|_p\|1\|_q$. When $q<\infty$, we have $\|1\|_q<\infty$ since $m(K)<\infty$.

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