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If $X$ is a finite measure space then one can show that if $1 \le p < q$ then $L^q \subseteq L^p$.

Is there anything known about the inclusion if $X$ is an arbitrary measure space? Or given some $p_1, \dots, p_n$ and $q_1,\dots, q_m$ can one always find a function $f$ with $f \in L^{p_i}$ but $f \notin L^{q_i}$?

The question I am now most interested in is: Does there exist a measure space such that for any arbitrary collection $p_i \in \mathbb N$ and $q_i \in \mathbb N$ (both not necessarily finite or countable) one can find $f$ such that $f \in L^{p_i}$ and $f \notin L^{q_i}$?

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newb
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  • Yes, this question came up after I read this thread. – newb Mar 19 '14 at 16:53
  • But it does not seem to contain the answer to my question. – newb Mar 19 '14 at 16:54
  • There is a necessary and sufficient condition on the measure contained in the answers of the thread. – Davide Giraudo Mar 19 '14 at 19:52
  • @DavideGiraudo Sorry I don't understand. The other answer contains the statement that if the space contains sets of arbitrarily small measure then it does not necessarily hold that $p < q$ implies $L^p \subset L^q$. Can you please elaborate a bit why this answers my question? In particular, does it follow that given $p_i$ and $q_i$ there is $f $ with $f \in L^{p_i}$ and $f \notin L^{q_i}$? – newb Mar 20 '14 at 08:56

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