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Let $0 <p < q < \infty$. Why exactly does the inclusion $L_{\mathrm{loc}}^q(\mathbb{R}^n) \subseteq L_{\mathrm{loc}}^p(\mathbb{R}^n)$ hold?

For clarification:



marked as duplicate by user296602, Shailesh, Cyclohexanol., user99914, Hans Engler Aug 22 '16 at 1:43

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  • $\begingroup$ Holder's inequality. $\endgroup$ – user296602 Aug 21 '16 at 21:16
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    $\begingroup$ This might be similar math.stackexchange.com/questions/66029/… $\endgroup$ – iamvegan Aug 21 '16 at 21:21
  • $\begingroup$ Thanks a lot. I totally forgot the regularity of the Lebesgue measure. $\endgroup$ – TheGeekGreek Aug 28 '16 at 9:31