I know from Wikipedia that $\ell^p(S)$ is a special case of an $L^p$ space, namely it is the space $L^p(S, \mu)$, where $\mu$ is the counting measure on $S$.
There are also the $\ell^{\infty}$ and $L^{\infty}$ spaces. $\ell^{\infty}$ is defined as the space of bounded sequences with the sup norm. I read, also on Wikipedia, that this space $\ell^{\infty}$ is a special case of an $L^{\infty}$ space, namely it is the space $L^p(\mathbb N, \mu)$, where $\mu$ is the counting measure on $\mathbb N$.
One can consider a slight generalization of $\ell^{\infty}$: For any set $S$, consider the space $\ell^{\infty}(S)$ of all bounded functions $S\to k$ ($k$ either $\mathbb R$ or $\mathbb C$) equipped with the sup norm. Question: Can one also consider this space as a special case of $L^{\infty}(\cdots)$? It seems, Wikipedia doesn't answer this question, in fact, the Wikipedia page on "L-infinity" doesn't even consider the space $\ell^{\infty}(S)$.
Sure, on any set $S$ you have a counting measure $\#$. Then there are no nonempty null sets and thus the essential supremum (as in the definition of $L^\infty(S,\#)$) is just a supremum. In particular $\ell^\infty(S)=L^\infty(S,\#)$.