Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied?
$L^p$ and $L^q$ space inclusion
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functional-analysis
measure-theory
lebesgue-integral
lp-spaces
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0Do you want $L^q(X,\mathcal B,m)$ to be any subset of $L^p(X,\mathcal B, m)$, or a *proper* subset? Also did you mean $p \leq q$ or something else in the question? – 2011-09-20
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3http://en.wikipedia.org/wiki/Lp_space#Embeddings – 2011-09-20
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0You should also consider the counting measure on a finite set - what happens in this case? – 2011-09-20
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0The question has been effectively answered by the answers below in the case of finite spaces. For infinite spaces it may be interesting to read: http://math.stackexchange.com/questions/55170/is-it-possible-for-a-function-to-be-in-lp-for-only-one-p – 2011-09-20
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0Related: http://math.stackexchange.com/questions/1371017/are-there-relations-between-elements-of-lp-spaces/1371051#1371051 – 2016-06-06