For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.

$L^p$ spaces are defined for $p\in(0,\infty]$ as follows. Let $(X,\mathcal F,\mu)$ be a measure space. For $p$ with $0 < p<\infty$ we write $L^p(X,\mu)$ ($L^p(X)$ or $L^p$ when there is no ambiguity), for the vector space of equivalence classes (for equality almost everywhere) of measurable functions $f$ such that $\int_X|f|^pd\mu$ is finite. When $1\leq p<\infty$ we endow $L^p$ with the norm $\lVert f\rVert_p:=\left(\int_X|f(x)|^pd\mu(x)\right)^{1/p}$, while for $0 < p < 1$ we write $\lVert f\rVert_p:=\int_X|f(x)|^pd\mu(x)$, which induces a metric on $L^p$.

For $p=+\infty$, $L^\infty$ is the space of equivalence classes of functions $f$ such that we can find a constant $C$ with $|f(x)|\leq C$ almost everywhere. Then $\lVert f\rVert_{\infty}$ is the infimum of constants $C$ satisfying the latter property, and is called the essential supremum of $|f|$.

These spaces are sometimes called Lebesgue spaces. If we work with counting measure on, for example, the set $\mathbb N$, we get sequence spaces $\ell^p$ and $\ell^\infty$ as special cases.