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  "body": "\n## Definition\n\nA *Young function* is a [[convex function|07Q7CNM5WF]] $\\Phi \\colon [0, \\infty) \\to [0, \\infty)$ with $\\Phi(0) = 0$ and $\\Phi(t) \\to \\infty$ as $t \\to \\infty$.\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $\\Phi$ be a Young function.\nFor a real or complex [[measurable function|YYNCJSR0PJ]] $f$ on $X$, the *Luxemburg norm* is\n$$\\norm{f}_\\Phi = \\inf \\set{\\lambda \u003e 0 : \\int_X \\Phi\\!\\left( \\frac{\\abs{f}}{\\lambda} \\right) d\\mu \\le 1},$$\nwith $\\inf \\emptyset = \\infty$.\nThe *Orlicz space* $L^\\Phi(\\mu)$ is the set of measurable $f$ with $\\norm{f}_\\Phi \u003c \\infty$, modulo equality [[almost everywhere|RARTMA0ZEC]].\n\n## Remarks\n\nExample: for $1 \\le p \u003c \\infty$, $\\Phi(t) = t^p$ is a Young function.\nThe condition $\\int (\\abs{f}/\\lambda)^p \\, d\\mu \\le 1$ means $\\lambda \\ge \\norm{f}_p$.\nSo $L^\\Phi = L^p$ with $\\norm{f}_\\Phi = \\norm{f}_p$.\n\nNon-example: $\\Phi(t) = \\sqrt{t}$ is not a Young function, because it is not convex.\nOn $[0, 1]$ with Lebesgue measure the Luxemburg functional is then $\\left( \\int \\abs{f}^{1/2} \\, d\\mu \\right)^2$.\nIt fails the triangle inequality: the indicator functions of $[0, \\tfrac12]$ and $(\\tfrac12, 1]$ each get the value $\\tfrac14$, but their sum gets the value $1$.\n\nA function $f$ lies in $L^\\Phi$ exactly when $\\int_X \\Phi(\\abs{f}/\\lambda) \\, d\\mu \u003c \\infty$ for some $\\lambda \u003e 0$.\nWith the Luxemburg norm, $L^\\Phi(\\mu)$ is a Banach space.\nSome authors allow Young functions with values in $[0, \\infty]$; this includes $L^\\infty$ as an Orlicz space.\n",
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