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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]], let $\\mathbb{K}$ be $\\R$ or $\\C$, and let $1 \\le p \u003c \\infty$.\nLet $\\mathcal{L}^p(\\mu)$ be the set of [[measurable functions|YYNCJSR0PJ]] $f \\colon X \\to \\mathbb{K}$ with $\\int_X \\abs{f}^p \\, d\\mu \u003c \\infty$, an [[integral of a non-negative function|DG76G5Z6XW]].\nCall $f \\sim g$ when $f = g$ [[almost everywhere|RARTMA0ZEC]].\nThe space $L^p(\\mu)$ is the set of [[equivalence classes|082RMRSY26]] of $\\mathcal{L}^p(\\mu)$ under $\\sim$, with pointwise operations on representatives.\nIts [[norm|08NYZ2FS6B]] is\n$$\\norm{f}_p = \\Big( \\int_X \\abs{f}^p \\, d\\mu \\Big)^{1/p}.$$\n\n## Remarks\n\nExample: on $\\R$ with Lebesgue measure, $f(x) = e^{-x^2}$ lies in $L^p$ for every $p$, since $\\int e^{-p x^2} \\, dx = \\sqrt{\\pi / p} \u003c \\infty$.\n\nNon-example: $g(x) = 1 / (1 + \\abs{x})$ on $\\R$ is in $L^2$ but not in $L^1$.\nIndeed $\\int_\\R (1 + \\abs{x})^{-1} \\, dx = \\infty$, while $\\int_\\R (1 + \\abs{x})^{-2} \\, dx = 2$.\n\nThe operations and the norm do not depend on the representatives.\nThe map $\\norm{\\cdot}_p$ is a norm by the [[Minkowski inequality|N4QZS912MT]], and $\\norm{f}_p = 0$ forces $f = 0$ almost everywhere.\nWith this norm $L^p(\\mu)$ is a Banach space, by the [[Riesz–Fischer theorem|WVV4MFB2SE]].\nFor $p = \\infty$ one uses the essential supremum instead, which gives the space [[L-infinity|ZT01G7BTSV]].\nWith the counting measure on $\\N$, $L^p$ is the sequence space $\\ell^p$.\n",
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