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Let $(X,M,\mu)$ be a measure space with $\mu(X) < 1$. Show that for any $1\le p<q$, we have $$L^q (X,\mu)\subset L^p(X,\mu).$$ Let $\ell^p(Z)$ denote the $L^p$ space of the integers equipped with the counting measure. Show that $$\ell^p(Z)\subset \ell^q(Z)$$ for any $1\le p < q$.


Okay, first Part of the problem already has been discussed and been answered by Davide Giraudo as following:$$ $$

let $1\leq p<q<\infty$ and $f\in L^q$. We put $E_n:= \left\{x\in X: \frac 1{n+1}\leq |f(x)|\leq\frac 1n\right\}$ for $n\in\mathbb N^*$. The sets $\{E_n\}$ are pairwise disjoint and by $2$ we get $\displaystyle\sum_{n=1}^{\infty} m(E_n)<\infty$. We have \begin{align*} \int_X |f|^pdm &=\int_{\{|f|\geq 1\}}|f|^pdm+\sum_{n=1}^{+\infty}\int _{E_n}|f|^pdm\\\ &\leq\int_X |f|^qdm+\sum_{n=1}^{+\infty}\frac 1{n^p}m(E_n)\\\ &\leq \int_X |f|^qdm+\sum_{n=1}^{+\infty}m(E_n)<\infty. \end{align*} Now we look at the case $q=+\infty$. If $m(E)<\infty$, since for each $f\in L^q$ we can find $C_f$ such that $|f|\leq C_f$ almost everywhere, we can see $f\in L^p$ for all $p$. Controversely, if $L^{\infty}\subset L^p$ for a $p<\infty$, then the function $f=1$ is in $L^p$ and we should have $m(E)<\infty$.

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marked as duplicate by Norbert, Did, Berci, icurays1, vadim123 May 5 '13 at 15:50

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