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Let $(X,\mathcal F,\mu)$ be a measure space.

For $p$ with $0 < p<\infty$ we write $L^p(X,\mu)$ (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 $\lVert f\rVert:=\left(\int_X|f(x)|^pd\mu(x)\right)^{1/p}$, while for $0<p<1$ we write $\lVert f\rVert:=\int_X|f(x)|^pd\mu(x)$, 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, this is known as 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.

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