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I was wondering relations of $L_p$ spaces..

Let $E$ be a measurable set. If $E$ is of finite measure, then $L_p(E) \subset L_q(E)$, $1 \le p \le q \le \infty$.

However, does it still hold if $E$ is infinite?

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It may hold, (when we take a countable set with counting measure) or not ($\Bbb R$ with Lebesgue measure). See math.stackexchange.com/questions/66029/… for information about inclusion of $L^p$ spaces. –  Davide Giraudo Oct 16 '12 at 15:36
    
I think the inclusion should be in the other direction. –  Your Ad Here Oct 16 '12 at 15:38
    
Your inclusion goes the wrong way. In finite measure spaces for $p \le q$ you have $L^p \supseteq L^q$. If you have a measure with no sets of arbitrarily small positive measure (like the counting measure) then your inequality is true. In general measure spaces none of these inclusions hold in general. –  Lukas Geyer Oct 16 '12 at 15:39
    
Thank you guys for your answers. –  Jon Lee Oct 18 '12 at 22:51

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