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My professor said that $\ell^p$ spaces are $L^p$ spaces with a discrete measure. But how can that be true if that the inclusion of the spaces is in different directions in the two cases? In the first case, $\ell^p \subset \ell^q$ if $p < q$, while $L^q \subset L^q$ if $p < q$!

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    Both the inclusions you mentioned are true only if some conditions are met. Check this link : http://math.stackexchange.com/questions/66029/lp-and-lq-space-inclusion – 2017-01-26
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    You write $L^p$, but what is the domain of the functions in that space? – 2017-01-26
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    We only have $L^q\subset L^p$ for $q$\ell^p$. – 2017-01-26
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    To elaborate Mariano's comment: The spaces $L^p[0,1]$ and $L^p(\mathbb{R})$ are different. – 2017-01-26
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    @MarianoSuárez-Álvarez Thanks, your comment made me realize I was being totally imprecise in my reasoning. – 2017-01-28

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The proof of the inclusion $L^q(\mu) \subset L^p(\mu)$ for $pfinite measure. This inclusion does not hold in general (consider $\mu$ the Lebesgue measure on $\mathbb{R}$).

What your professor was hinting towards was the fact that $\ell^p=L^p(\mu)$, with $\mu$ the measure on $\mathbb{N}$ (with $\sigma$-algebra the power set of $\mathbb{N}$, hence the term discrete measure) given by $$\mu(A)=|A|,$$ where $|A|$ denotes the number of elements of $A$.