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I am trying to prove, the following question

$ 1\leq p <r <q< \infty $ . Prove that $L^p \cap L^q \subseteq L^r $

The prove that I have written for this is rather straightforward,

basically

$$ |f|^p < |f|^r < |f|^q , $$ which implies that

$$ \int |f|^pdu < \int |f|^rdu < \int|f|^qdu $$

My question is that when moving to the integral sign in step two do I need to invoke any integration theorem such as Lebesgue Dominated convergence theorem , or Lebesgue Increasing Convergence theorem ?

If so , then why and how ?

Edit: I just realised that this proof is wrong as obviously my first inequality doesn't hold for cases where $f<1$

So now I am thinking of proving this with holder's inequality

Joe
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Noob101
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    It's not true that $||f||p < ||f||_r < ||f||_q$ so you're trying to prove a false, but stronger statement. You should break $f$ up into $f\chi{{|f| > 1}}$ and $f\chi_{{|f| \le 1}}$ – mathworker21 May 09 '17 at 06:27
  • @mathworker21 It is true , Infact I found a proof on the stackexchange. https://math.stackexchange.com/questions/235173/liapunovs-inequality-for-l-p-spaces – Noob101 May 09 '17 at 06:30
  • That's very different from $||f||_p < ||f||_r < ||f||_q$. For an example, take $\frac{1}{x}$ on $[1,2]$. – mathworker21 May 09 '17 at 06:32

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