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## Main Question or Discussion Point

Suppose [tex]0 < p < q < \infty[/tex]. Then [tex]L^p \nsubseteq L^q[/tex] iff [tex]X[/tex] contains sets of arbitrarily small positive measure.

I have proved one part, namely, if [tex]X[/tex] contains sets of arbitrarily small positive measure then [tex]L^p \nsubseteq L^q[/tex]

Can anyone give some hints to solve the other part?

Thanks

I have proved one part, namely, if [tex]X[/tex] contains sets of arbitrarily small positive measure then [tex]L^p \nsubseteq L^q[/tex]

Can anyone give some hints to solve the other part?

Thanks