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  "body": "\n## Summary\n\nFix a convex growth function $\\Phi$.\nThe Orlicz space $L^\\Phi(\\mu)$ consists of the measurable functions $f$ for which $\\Phi(\\abs{f})$ is integrable after scaling $f$.\nThe choice $\\Phi(t) = t^p$ gives the [[Lp space|SASP14MA1Z]] $L^p$, so Orlicz spaces generalise the $L^p$ scale.\nThey are Banach spaces with the Luxemburg norm.\nThey describe integrability conditions that lie between the $L^p$ spaces, such as $L \\log L$, which appears in the theory of maximal functions and singular integrals.\n\n## Explanations\n\n### L1 human\n\nThe space $L^p$ asks that $\\abs{f}^p$ have finite integral.\nAn Orlicz space replaces the power $t^p$ by another convex function $\\Phi(t)$.\nWith $\\Phi(t) = e^t - 1$ one asks for exponential integrability, which is stronger than every $L^p$ condition on a finite measure space.\nWith $\\Phi(t) = t \\log(1 + t)$ one asks, on a finite measure space, for a little more than integrability, but less than any $L^p$ with $p \u003e 1$.\n",
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