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Suppose $1 \le p < q < \infty$ and $(X,\mu)$ is a Lebesgue measure space. Also suppose $X$ is of finite measure. Prove that $L^q \subset L^p$.

First, we use Holder's inequality and find $$\int_X |f|^p \, dx \le \left( \int_X |f|^q \, dx \right)^{\frac pq} \left(\int_X \, dx \right)^{1-\frac pq}$$ which reduces to $$\|f\|_p \le \|f\|_q \mu(X)^{\frac 1p - \frac 1q} = C\|f\|_q$$ Does this give me the conclusion $L^q \subset L^p$?

user265141
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1 Answers1

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Elements of $L^q$ are precisely those classes of measurable functions with $\| f \| _q < \infty$. Now if you have such an $f \in L^q$, due to the inequality you have found you will also have $\| f \| _p \le C \| f \| _q < \infty$, so $f \in L^p$, therefore $L^q \subset L^p$.

Alex M.
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  • but you need $\mu(X)$ to be finite. Don't you? –  Aug 28 '15 at 16:09
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    @zap: It is already assumed in the statement of the problem, read it carefully: "Also, suppose $X$ is of finite measure". You might also want to delete your answer. – Alex M. Aug 28 '15 at 16:11
  • you're right. I didn't read this part. Or maybe was edited –  Aug 28 '15 at 16:12