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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $f \\colon X \\to [-\\infty, \\infty]$ be a [[measurable function|YYNCJSR0PJ]].\nA number $c \\in [-\\infty, \\infty]$ is an *essential upper bound* of $f$ if the set $\\set{x \\in X : f(x) \u003e c}$ is a [[null set|VQQM4Y550T]].\nThe *essential supremum* of $f$ is the [[infimum|04KFTA34C5]] of its essential upper bounds in the [[extended real line|0GSZ7B0CGR]]:\n$$\\operatorname{ess\\,sup} f = \\inf \\set{c \\in [-\\infty, \\infty] : \\mu(\\set{x \\in X : f(x) \u003e c}) = 0}.$$\n\n## Remarks\n\nExample: on $\\R$ with Lebesgue measure, let $f = \\mathbf{1}_{\\Q}$, the indicator function of the rationals.\nThe set $\\set{f \u003e 0} = \\Q$ is countable, so it has measure $0$, and $\\operatorname{ess\\,sup} f \\le 0$.\nFor $c \u003c 0$ the set $\\set{f \u003e c}$ is all of $\\R$, so $\\operatorname{ess\\,sup} f = 0$, while $\\sup f = 1$.\n\nNon-example: the number $1/2$ is not an essential upper bound of $g(x) = x$ on $[0, 1]$ with Lebesgue measure.\nThe set $\\set{g \u003e 1/2} = (1/2, 1]$ has measure $1/2 \\ne 0$.\nIn fact $\\operatorname{ess\\,sup} g = 1$.\n\nThe essential supremum is itself an essential upper bound, since $\\set{f \u003e c}$ is a countable union of the sets $\\set{f \u003e c + 1/n}$.\nThe *essential infimum* is $\\operatorname{ess\\,inf} f = -\\operatorname{ess\\,sup}(-f)$.\nThe quantity $\\operatorname{ess\\,sup} \\abs{f}$ is the norm $\\norm{f}_\\infty$ of the space [[L-infinity|ZT01G7BTSV]].\n",
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