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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]], and let $\\mathbb{K}$ be $\\R$ or $\\C$.\nA [[measurable function|YYNCJSR0PJ]] $f \\colon X \\to \\mathbb{K}$ is *essentially bounded* if there is $M \\ge 0$ with $\\mu(\\set{x \\in X : \\abs{f(x)} \u003e M}) = 0$.\nIts norm is the [[essential supremum|WPC2XRMB9Y]]\n$$\\norm{f}_\\infty = \\inf \\set{M \\ge 0 : \\mu(\\set{x : \\abs{f(x)} \u003e M}) = 0}.$$\nThe space $L^\\infty(\\mu)$ is the set of essentially bounded measurable functions modulo equality [[almost everywhere|RARTMA0ZEC]], with the norm $\\norm{\\cdot}_\\infty$.\n\n## Remarks\n\nExample: on $[0, 1]$ with Lebesgue measure, let $f(x) = x$ for rational $x$ and $f(x) = 0$ otherwise.\nThe set where $f \\ne 0$ is countable, so it has measure $0$, and $\\norm{f}_\\infty = 0$.\nSo $f$ is the zero element of $L^\\infty$.\n\nNon-example: $f(x) = 1/x$ on $(0, 1]$ is not essentially bounded.\nFor each $M \u003e 0$, the set $\\set{f \u003e M}$ contains the interval $(0, \\min(1, 1/M))$, which has positive measure.\n\nThe infimum in the definition is attained: $\\abs{f} \\le \\norm{f}_\\infty$ almost everywhere.\nWith the counting measure on a set $I$, $L^\\infty$ is the space $\\ell^\\infty(I)$ of bounded families with the supremum norm.\n",
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