Embeddings $L_p$ in $L_q$ for $p<q$.

Embeddings $L_p$ in $L_q$ for $p<q$. [closed]

Let (X.S,m) be measure space and $p < q$.

I would like the references or idea of the proof of the followig result:

Lp(X, m) is contained in Lq(X, m) iff S does not contain sets of arbitrarily small non-zero measure.

Thanks.

edited Dec 12 2011 at 0:54

Mark Sapir
27.6k●2●57●118

asked Dec 11 2011 at 23:04

arc
197●3

Doesn't that condition force the measure to be atomic? and then the atomic case does not seem too hard – Yemon Choi Dec 11 2011 at 23:15

This is too elementary for MO, which is devoted to research level questions, so I vote to close. You might get a response at math.stackexchange.com, but read their faq page before posting there. – Bill Johnson Dec 12 2011 at 5:35

closed as too localized by Bill Johnson, Alain Valette, Matthew Daws, Zev Chonoles, Dmitri Pavlov Dec 12 2011 at 11:33

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