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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]], and let $P(x)$ be a statement about the points $x \\in X$.\nThe statement $P$ holds *$\\mu$-almost everywhere* if there is a [[null set|VQQM4Y550T]] $N \\in \\Sigma$ such that $P(x)$ is true for every $x \\in X \\setminus N$.\n\n## Remarks\n\nLet $\\lambda$ be Lebesgue measure on $\\R$ and let $f(x) = 1$ for rational $x$ and $f(x) = 0$ otherwise.\nThen $f = 0$ holds $\\lambda$-almost everywhere: take $N = \\Q$, a countable set, so $\\lambda(\\Q) = 0$.\nThe statement $x \\ne 0 \\Rightarrow x \u003c 0$ does not hold $\\lambda$-almost everywhere, since it fails on $(0, \\infty)$, and every measurable set containing $(0, \\infty)$ has measure $\\infty$.\nThe set where $P$ fails is only required to lie inside a null set; it need not be measurable itself.\nThe usual abbreviation is a.e., or $\\mu$-a.e.\nOn a probability space, almost everywhere is called [[almost surely|0ST21PGDAP]].\n",
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