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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 $$\int_X|f|^p\mathrm{d}\mu$$ is finite.

When $$1\leq p<\infty$$ we endow $$L^p$$ with the norm $$\lVert f\rVert_p:=\left(\int_X|f(x)|^p\mathrm{d}\mu(x)\right)^{1/p}$$ When $$0 < p < 1$$ we write $$\lVert f\rVert_p:=\int_X|f(x)|^p\mathrm{d}\mu(x)$$ 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.